Download Unit 3 Geometry: Understanding Congruence and Properties of Angles

Unit 3 Geometry: Understanding Congruence
and Properties of Angles
Introduction
In this unit, students will review the basic concepts of geometry, such as lines, rays, line
segments, angles, and polygons. They will use informal arguments to establish facts about the
sum of the angles in a triangle, the exterior angles in a triangle, and the angles created when
parallel lines are intersected by another line. Students will learn to make sketches and
recognize and create counterexamples. Students will also be introduced to the idea of
congruence, and they will recognize and use congruent triangles to produce informal
arguments.
Vocabulary. The most important terms, learned in this and previous grades, are as follows:
polygon
parallelogram
line segment
right angle
equilateral
vertical angles
supplementary angles
quadrilateral
trapezoid
line
acute
scalene
alternate angles
co-interior angles
pentagon
rhombus
ray
obtuse
isosceles
corresponding angles
hexagon
parallel
perpendicular
straight angles
congruent
exterior angles
To help students who are struggling with these terms, use the cards from BLM Geometric
Terms (pp. D-111–113). Students can match terms with the different representations of the
same concept. You could also use the cards to play a memory game: Player 1 lays out the
cards face down and turns over pairs at random. If the pair matches, Player 1 collects the pair; if
not, Player 1 turns the cards face down again. Player 2 then takes a turn. You might also want
to use this BLM as a summary sheet.
Trapezoids and parallelograms. There are two possible definitions of a trapezoid:
1. A quadrilateral with exactly one pair of parallel sides
2. A quadrilateral with at least one pair of parallel sides
When using the first definition, parallelograms are different from trapezoids, but when using the
second definition, all parallelograms are trapezoids. Both definitions are legitimate; we use the
first definition.
Protractors. Students will need to use protractors in this unit. If you need additional protractors for
individual students, photocopy a protractor (or BLM Protractors, p. D-114) onto a transparency
and cut it out. These protractors are also convenient to use on an overhead projector.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-1
Grids. Students will need to use grid paper throughout this unit. In addition, you will need a predrawn grid on the board for some lessons. If you do not have such a grid, photocopy BLM 1 cm
Grid Paper (p. I-1) onto a transparency and project it onto the board. This will allow you to draw
and erase shapes and lines on the grid without erasing the grid itself. If students are not working
in grid paper notebooks, provide them with grid paper or BLM 1 cm Grid Paper.
Paper folding. Many activities in these lessons involve paper folding to make, check, and
examine shapes and angles. Unless otherwise noted, the starting shape is a regular 8 1/2" by
11" sheet of paper. Sometimes the starting shape is a hand-drawn circle to make sure there are
no angles for students to start with or refer to.
Technology: dynamic geometry software. Students are expected to use dynamic geometry
software to draw geometric shapes. Some of the activities in this unit use a program called The
Geometer’s Sketchpad®, and some are instructional—they help you teach students how to use
the program. If you are not familiar with The Geometer’s Sketchpad®, the built-in Help Center
provides explicit instructions for many constructions. Use phrases such as “How to construct
congruent angles” or “How to construct a line segment of given length” to search the Index.
NOTE: If you use different dynamic geometry software to complete these activities, you may
need to adjust the instructions provided.
Summary BLMs. Some crucial definitions, properties of shapes, and step-by-step instructions
used in this unit are summarized on BLMs, for easy reference. The chart below lists the
summary BLMs and the topics they cover. Summary BLMs are noted in some specific lessons,
but you can use them at any point where they will support students.
BLM Title
Topics Covered
Geometric Terms
(pp. D-111–113)
Geometric terms regarding shapes, lines, angles, properties,
and congruence—use as a summary chart, for matching, or a
memory game
Using Protractors
(Summary)
(p. D-115)
Measuring an angle, drawing an angle, drawing a line perpendicular
to a given line through a point, and drawing a line parallel to a given
line through a point
Angle Properties
(Summary)
(p. D-133)
Supplementary, corresponding, co-interior, alternate, and
vertical angles
Properties of Angles in a
Triangle (Summary)
(p. D-134)
Sum of the angles in a triangle, exterior angle in a triangle, angles in
an isosceles triangle, and congruence rules
D-2
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-1
Points and Lines
Pages 72–73
Standards: preparation for 8.G.A.5
Goals:
Students will review the basic geometric concepts and notation for points, lines, line segments,
and rays.
Students will review intersecting lines and line segments.
Prior Knowledge Required:
Can use a ruler to draw lines
Can measure distances to the nearest millimeter
Vocabulary: endpoint, intersect, intersection point, line, line segment, point, ray
Materials:
The Geometer’s Sketchpad®
Review the concepts of a point, line, line segment, and ray. Draw a dot on the board.
Explain that the dot represents a point. SAY: A point is an exact location. A point has no
size—no length, width, or height. The dot has size, or you couldn’t see it, but real points do not.
Draw on the board:
line
Explain that a line extends in a straight path forever in two directions. It has no ends. To show a
line that extends forever in both directions, we draw arrows at both ends of the line. Lines that
we draw have a thickness, but real lines do not.
Draw on the board:
line segment
Explain that a line segment is the part of a line between two points, called endpoints. The
endpoints are usually shown as dots. A line segment has a length that can be measured, but it
has no width or thickness.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-3
Draw on the board:
ray
Explain that a ray is part of a line that has one endpoint and extends forever in the other
direction. Similar to a line, it has no length, width, or thickness.
Write on the board:
line segment
line
ray
On another part of the board, draw a line segment and ask students to signal whether it is a line
(by pointing to the left), a line segment (by pointing up), or a ray (by pointing right). Change the
endpoints so that the object becomes first a ray, then a line, each time asking students to signal
the correct term. Repeat with lines, rays, and line segments in different positions and with
varying lengths.
Review naming a point, line, ray, and line segment. Explain that we use capital letters to
name points. Draw a point on the board, and write “A” beside it. Then draw a line through
point A and mark another point on the line. Label the second point “B,” as shown below:
B
A
SAY: To name a line, we give the names of any two points on the line. This is line AB. The order
of the endpoints does not matter, so we can also name this line BA. Write on the board:
line AB or BA
Draw on the board:
C
E
D
SAY: Since I labeled three points on this line, we can name this line in many different ways. You
can use any two points. Ask students to copy the diagram and then give all possible names for
the line. (CD, DC, DE, ED, CE, EC)
Draw on the board:
F
D-4
G
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Explain that, since a ray always has an endpoint, we have to use that point when naming the
ray and write that point first. So this is ray FG, but not ray GF.
Exercises: Which point is the endpoint of the ray? Name the ray in all possible ways.
a)
b)
Q
H
J
R
P
M
Answers: a) endpoint H, ray HJ or HM; b) endpoint R, ray RP or RQ
Draw a line segment on the board. Explain that a line segment has two very clear endpoints,
so to name a line segment you have to name both endpoints. The order in which you name the
endpoints does not matter. Have students suggest labels for the endpoints and write both
possible names for the line segment. (e.g., KL and LK)
Draw on the board:
N
S
O
SAY: This diagram shows three different line segments. A line segment is the part of the line
between the endpoints, so even though all three points are on the same line, the line segments
are different. Trace the line segment NO with your finger and ask students to name it. (NO or
ON) Repeat with the other two line segments. (OS or SO, NS or SN)
Exercises: Name the lines, line segments, and rays in the diagram in all possible ways.
Say whether each object is a line, a ray, or a line segment.
a)
b)
Y
U
Z
V
W
X
A
Answers: a) line segment UW or WU, ray WV; b) line YZ or ZY, line segment AZ or ZA, ray YX
Introduce the intersection point. Tell students that an intersection point is a point that lines,
line segments, or rays have in common. Draw on the board:
Indicate the intersection point in each picture. Explain that, if the location of a point is clear,
you do not need to draw a dot to show it. When two lines, rays, or line segments intersect
(meet or cross), there is only one intersection point, so there cannot be any doubt about its
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-5
location. So there is no need to draw a dot in the pictures on the board. Remove the dots from
the intersection points.
Draw on the board:
A
B
D
C
ASK: Does AB intersect CD? Explain that the answer depends on whether AB and CD are lines,
line segments, or rays. As a class, check all possible combinations of lines and line segments,
as shown below:
A
A
B
D
C
A
B
D
C
A
B
D
C
B
D
C
Extend the lines to show intersection. (if either AB or CD is a line segment, they do not intersect;
if both are lines, they intersect) ASK: Why is there no intersection when either one is a line
segment? (because the positions of AB and CD mirror each other; when we make one a line
and the other a line segment, if they don’t intersect in one case, they won’t intersect in the mirror
of the case)
Explain that the situation with rays is more complicated; the answer depends on which side of
AB can be extended. SAY: Let’s suppose that the object AB or BA is a ray. Remind students
that a ray is named with the endpoint first. Ask students to check all possible cases, as
shown below:
1.
2.
A
A
B
D
C
3.
5.
C
A
B
D
6.
A
C
B
D
4.
A
B
D
C
A
C
B
D
D
C
7.
B
8.
A
C
B
D
A
C
B
D
(1. do not intersect, 2. do not intersect, 3. do not intersect, 4. do not intersect, 5. do not
intersect, 6. do not intersect, 7. intersect, 8. intersect)
D-6
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Activities 1–2
Use The Geometer’s Sketchpad® for the following two activities.
(MP.5) 1. Proper use of Point and Line tools. Show students how to draw lines, line
segments, and rays using the Line tool of The Geometer’s Sketchpad®. Then go through the
following tasks and questions:
a) Ask students to draw a line and an independent point. Ask them to move the point so that it
looks like it is on the line. Then have them move or turn the line. ASK: Does the point stay on
the line? (no)
b) Ask students to draw a point. Then have them draw a line using that point as one of the
points you draw the line through. Note that the point is highlighted when you select it; this will
ensure the line passes through the point. Have students move the points and the line itself.
ASK: Does the line move? (yes) Does it still pass through the point you started with? (yes)
c) Show students how to mark an intersection point of two lines. Draw two lines so that they
intersect. Start drawing a point and ASK: How can you tell that the point is on both lines?
(both lines get highlighted) Have students construct two lines and mark the intersection point.
Have them move the lines and point out that the intersection point remains on both lines.
(MP.5) 2. Length of a line segment.
a) Have students draw a line segment and use the Measure menu options to find its length.
Then ask them to try to move the endpoints so that the length of the segment becomes 3 cm.
ASK: Is it easy or hard to do? (hard) If you move the line segment around, does its length
change? (no) If you move the endpoints around, does the length of the line segment change?
(yes) Show students how to draw a line segment that is 3 cm long by following these steps:
Step 1: Use the Number menu options to create a new parameter. Change the value to 3 and
mark it as “distance.”
Step 2: Mark a point. It will be one endpoint of the line segment. Select the point and
the parameter.
Step 3: Using the Construct menu options, construct a circle (by center and radius).
Step 4: Use the Line Segment tool to create a line segment between the center of the circle and
any point on the circle. Hide the circle using the right-click menu options.
ASK: Will moving the endpoints change the length of the line segment now? (no)
b) Have students draw a line segment and label it AB. Challenge students to draw a line
segment CD that is exactly the same length as AB. Discuss the two options:
1. Use the copy and paste option to create a copy of the line segment at a given point. When
you modify the original line segment, the copy does not change.
2. Measure the length of AB. Use the length of AB as the parameter in the steps in part a).
Modifying the line segment AB modifies the second line segment accordingly. This method will
be more useful when students work with congruent triangles.
(end of activities)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-7
Extensions
1. a) Draw two points. Label them A and B.
b) Draw the line AB.
c) Draw line CD so that point A is not on CD, but point B is on CD.
d) Do lines AB and CD intersect? If yes, what is the intersection point?
Answers: d) lines AB and CD intersect, B is the intersection point
(MP.3) 2. a) Jen thinks rays DE and DF are the same ray. Is she correct? Explain.
F
E
D
b) Name one point that is on ray DF but not on ray EF.
c) Ron thinks rays ED and ray FD are the same ray. Is he correct? Explain.
Answers: a) yes, the rays are part of the same line, pointing in the same direction, and with the
same starting point, so they are the same ray; b) point D on the ray DF is not on ray EF; c) no,
point F is on ray FD, but not on ray ED, so these are not the same ray
D-8
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-2
Angles and Shapes
Pages 74–75
Standards: preparation for 8.G.A.5
Goals:
Students will review naming angles and polygons.
Prior Knowledge Required:
Knows what an angle is
Can identify polygons
Understands clockwise and counter-clockwise
Vocabulary: arc, arms, endpoint, intersect, intersection point, polygon, right angle,
straight angle, vertex, vertices
Materials:
The Geometer’s Sketchpad®
Review angles. Remind students that, in mathematics, when two rays have a common
endpoint, they make an angle. Draw two rays with a common endpoint. Draw the pictures below
on the board. For each one, have students signal thumbs up if the picture shows an angle and
thumbs down if it does not.
Exercises: Does the picture show an angle?
a)
b)
c)
d)
f)
e)
Answers: a) no, b) yes, c) no, d) yes, e) yes, f) no
Remind students that the rays are called the arms of the angle, and the common endpoint is
called the vertex of the angle. SAY: The plural of vertex is vertices. Have students each draw a
few angles, exchange notebooks with a partner, and identify the vertex and extend the arms of
the angles drawn by their partner.
Explain that, when the vertex of an angle is a clear point, such as a common endpoint of two
rays, you do not need to draw a dot to show the vertex. Erase the dots in the angles in parts b),
d), and e) from the previous exercises, and explain that these are still pictures of angles.
Naming angles. Explain that, to name an angle, you need to name three points: one on each
arm and the vertex. SAY: The vertex is always the middle letter. Draw the picture on the next
page and explain that you can name this angle in two ways: XYZ or ZYX, but not YXZ. SAY: To
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-9
make it clear that you are naming an angle and not anything else, such as a triangle, we use the
angle sign “” before the letters. Write the two names for the angle underneath the picture.
XYZ or ZYX
Exercises: Name the angle in two ways.
a)
b)
C
L
K
B
c)
T
M
A
E
N
Answers: a) CAB, BAC; b) KLM, MLK; c) TEN, NET
Remind students that when there are several points on a ray, the ray can be named in several
ways; you just need to name the endpoint of the ray first. Add a point D between points A and B
in the angle in part a) of the previous exercises to illustrate the point, as shown below:
C
A
D
B
Trace ray AB and SAY: We can name this ray AB or AD. This means that we can name this angle
in more than two ways: CAB, BAC, CAD, and DAC. Write all four names on the board.
Exercises:
1. Name the angle in all possible ways.
Q
Answers: AQB, BQA, AQE, EQA, KQB, BQK, KQE, EQK
2. a) Which of the following are possible names for this angle?
CAT, CTP, PTA, UTA, UTC, TCP, PTU
b) Write three more different names for the angle.
Answers: a) CTP, PTA, UTA, UTC; b) any three of CTU, ATP, PTC, ATU
D-10
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Naming angles that are parts of other angles. Draw on the board:
Explain that point A in this picture is the vertex of several angles. For example, angle BAD is the
angle of the rectangle, but angle CAD also has vertex A, angle BAC has vertex A, and both
angles are parts of the angle BAD. SAY: We often use arcs, a part of a circle around the vertex
of an angle, to show angles that are part of other angles. ASK: What is the vertex of the angle
marked with the arc in this picture? (A) Circle the letter A and the vertex itself. ASK: What angle
does the arc in the picture show? (CAD) To prompt students to see the answer, thicken the rays
AC and AD. You can also extend the rays and add arrows beyond the points C and D, as shown
below. Then ask students to look for points on these rays. (C and D)
Exercises: Copy the picture. Circle the vertex of the angle with the arc. Draw rays along the line
segments the arc ends at. Then name the angle marked by the arc.
a) A
b) H
c) M
B
G
N
D
C
E
F
P
R
Q
Bonus: Name 4 more angles in each picture.
Answers: a) ADC or CDA, b) GHF or FHG, c) MNR or RNM, Bonus: answers will vary
Point out that in part c) of the previous exercises the arms of the angle MNR form a straight line.
SAY: We call such angles straight angles.
NOTE: To provide more practice to students who need it, mark another arc on each of the
pictures in the previous exercises, as shown in the following exercises, and have them identify
the new angles.
Exercises: Name the angle marked by the arc.
B
a) A
b) H
G
c)
M
N
D
C
E
F
P
Answers: a) DBC or CBD, b) GFE or EFG, c) MNQ or QNM
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
R
Q
D-11
Draw the picture below on the board and have students copy it in their notebooks:
ASK: Which angle in this diagram is the angle ACB? Have a volunteer mark it on the board.
Repeat with other angles, such as DCB, BAD, and BCA. Students can use different
numbers of arcs or different colors of arcs to mark the angles.
Naming an angle with one letter. Point out that sometimes, when there is no chance of
confusion, only the vertex letter is used to name an angle. Draw on the board:
Explain that, in this rectangle, there is only one possible D, but there are three possibilities for
A: BAD or BAC or DAC.
Review polygons. Remind students that polygons are flat shapes with sides that are line
segments, so the sides are straight. Sides of polygons only meet at vertices, and exactly two
sides meet at each vertex. Sides of polygons do not intersect except at endpoints. Draw on
the board:
polygons
not polygons
Naming polygons. Explain that, to name a polygon, you need to write the letters that name the
vertices in order. Draw a quadrilateral on the board and have students suggest names for
different vertices. Point out that the letters can be any from the alphabet and do not need to be
in alphabetical order, as in the example shown below:
SAY: To write a name for this polygon, you can start at any vertex. For example, I can start at F
and read the letters clockwise to name this polygon FQMS. Or I could read the letters
counterclockwise to name this polygon FSMQ. I cannot write M as the next letter after F.
Students can check each other’s work in the next exercises.
D-12
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Exercises: Name the polygon.
P
a) A
b) H
G
c)
d)
U
W
X
F
D
Q
C
J
E
V
Y
K
Sample Answers: a) APQD, b) CHE, c) JGFK, d) UWXYV
Activity
Use The Geometer’s Sketchpad® for this activity.
Using the Polygon tool. Show students how to use the Polygon tool to create polygons. Draw
students’ attention to the different options of the Polygon tool. Have them use the option that
makes the sides of the polygon visible and selectable. Teach students to label polygons. Have
them create triangle ABC, quadrilateral DEFG, and pentagon HIJKL.
Show students how to measure the length of the sides of polygons. Have them measure the
sides of the polygons they created.
(end of activity)
Extensions
1. Name all the triangles in the picture.
Answers: ABC, ACE, ECD, ACD
2. a) Draw line segment FG that is 6 cm long.
b) Mark point H on FG so that line segments FH and GH are the same length.
c) Draw a line that intersects FG at point H.
d) On the line you drew in part c), mark points J and K, so that HJ = HK = 3 cm.
e) Join the points F, K, G, and J to create a quadrilateral.
f) What type of quadrilateral have you produced?
g) Name all the triangles and the quadrilateral in your diagram.
Sample answers: a) to e) see diagram below; f) rectangle; g) triangles: FHK, FHJ, FGK, FGJ,
FKJ, GKJ, GKH, GJH, quadrilateral: FKGJ
K
F
G
H
J
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-13
G8-3
Perpendicular and Parallel Lines
Pages 76–78
Standards: preparation for 8.G.A.5
Goals:
Students will review perpendicular and parallel lines.
Students will review notation for perpendicular and parallel lines, equal line segments,
and right angles.
Students will review properties of special quadrilaterals, such as equal sides, parallel sides,
and right angles.
Prior Knowledge Required:
Can identify and name lines, line segments, and rays
Can draw and measure line segments
Can identify a right angle
Can name an angle and identify a named angle
Can identify and name polygons
Vocabulary: endpoint, hash marks, intersect, intersection point, parallel, parallelogram,
perpendiculars, quadrilateral, rhombus, square, trapezoid
Materials:
rectangular sheet of paper or a set square
rulers
BLM Geometric Terms (2) (p. D-112, optional)
The Geometer’s Sketchpad®
Review perpendicular lines. Explain that when two lines, rays, or line segments meet at a right
angle, we say that they are perpendicular. Draw the pictures in the following exercises on the
board and have students identify the perpendicular objects. Students can signal thumbs up for
objects that are perpendicular and thumbs down for objects that are not. Invite volunteers to
check whether the lines are perpendicular by using a corner of a page or a set square and mark
the right angles with a little square.
Exercises: Are the objects perpendicular?
a)
b)
c)
e)
g)
D-14
f)
d)
h)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Answers: a) yes, b) yes, c) no, d) yes, e) yes, f) no, g) no, h) yes
Ask students where they see perpendicular lines or line segments–also called perpendiculars–
in the environment. (sample answers: sides of windows and desks, intersections of streets, the
frame of a poster or painting)
Introduce the notation for perpendicular objects. Explain that there is a symbol people use
instead of the word “perpendicular.” Add labels to the pictures from parts a), e), and d) from the
previous exercises as shown below. Demonstrate on the first picture how to write the sentence
AB CD, which means that line AB is perpendicular to line CD. Have students do the rest of the
exercises individually.
Exercises:
1. Name the perpendicular lines, rays, or line segments.
a)
b)
c)
C
E
Bonus:
K
N
A
B
H
F
D
T
M
O
G
P
Answers: a) AB  CD; b) EF  FG; c) HK  HM; Bonus: TO  ON, TP  ON, PO  ON
2. Which sides look like they are perpendicular?
Answers: a) AB  BC, AB  AD, CB  CD, AD  CD; b) GH  FG; c) IM  LM, IM  JI;
d) ON  NR
A perpendicular through a point. Explain that sometimes we are interested in a line that
meets two conditions: it is perpendicular to a given line and passes through a given point.
SAY: Remember that a line segment is always part of a line. We might draw a line segment in a
picture, but the segment is part of a line. Draw on the board:
K
L
N
M
ASK: In this picture, which line segments are perpendicular to the line NM? (KN, LM)
What other line segment is here? (LN) Explain that, out of the three line segments, two are
perpendicular to the line NM and two pass through the point L, but only one line segment does
both: LM is perpendicular to the line NM and passes through the point L. SAY: line LM can be
called “the perpendicular to line NM through the point L.” There is no other line that satisfies
both conditions.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-15
Exercises: Identify …
a) the lines that are perpendicular to the line AB,
b) the lines that pass through point P, and
c) the line that satisfies both conditions.
I
Answers: a) i) EC, DH, ii) DK, EF; b) i) GF, DH, ii) DK, CI, AB; c) i) DH, ii) DK
Why perpendiculars are important. Discuss with students why perpendiculars are important
and where are they used in real life. For example, if construction workers need to cut wooden
floorboards into perfect rectangles, they can measure the necessary length on one side of the
board and make the cut at a right angle to the side of the board. In other words, they make a cut
perpendicular to the board side.
Review parallel lines. Remind students that parallel lines are straight lines that never intersect,
no matter how much they are extended. Ask students to think about where they see parallel
lines. (e.g., a double center line on a highway, the edges of construction beams)
Remind students how to mark parallel lines with the same number of arrows, as shown below:
Emphasize that the number of arrows and their direction has to match in each set of parallel
lines, but there is no rule about where the arrows should be pointing. Reverse the arrows on
sides BC and AD and explain that this picture still shows that the quadrilateral has two pairs of
parallel sides.
Exercises: Copy the diagram. Mark the sides that look parallel with the same number of arrows.
a)
b)
c)
d)
Answers:
a)
D-16
b)
c)
d)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Review the symbol for parallel lines (||). Return to the previous picture of parallelogram
ABCD. Explain that there is a symbol for parallel lines. Write “||” on the board. SAY: This symbol
resembles the double “l” in the word “parallel.” In the picture of parallelogram ABCD, line
segment AB is parallel to line segment CD, so we can write AB || CD. Write the expression on
the board. ASK: What other parallel line segments are in this picture? (AD and BC) Have a
volunteer write that on the board with the symbol.
Ask students to label the polygons they drew in the previous exercises with letters and write the
expressions showing which sides are parallel. (selected answer: KL || MN) Point out that since
in part d) there are three sides parallel to each other, and the expression only refers to two
sides, they will need to write each pair of three sides separately. (for shape OPQRSTU,
OP || UT, UT || SR, and SR || OP) Have students exchange notebooks with a partner to check
each other’s answers.
Review parallelograms and trapezoids. Remind students that quadrilaterals can be classified
by how many pairs of parallel sides they have. ASK: Can a quadrilateral have three parallel
sides? (no) Why not? (a quadrilateral has 4 sides, so 3 parallel sides would mean only 1 side
remains and it would not be able to close the shape) How many pairs of parallel sides can a
quadrilateral have? (0, 1, or 2) What do we call a quadrilateral that has exactly one pair of
parallel sides? (trapezoid) What do we call a quadrilateral that has two pairs of parallel sides?
(parallelogram) Invite volunteers to draw a few examples of each kind on the board and mark
the parallel sides. Point out that most quadrilaterals that have no parallel sides do not have a
special name.
Explain that in the next exercises the sides that look parallel are, indeed, parallel.
Exercises: Copy the quadrilateral. Mark the parallel sides with the same number of arrows.
Identify the quadrilateral as a parallelogram, trapezoid, or neither.
a)
b)
c)
d)
Answers:
a)
parallelogram
b)
c)
parallelogram
d)
neither
trapezoid
Review markings for equal sides. Remind students that we use hash marks to show equal
sides—in other words, to show that the sides are line segments of equal length. Similar to
parallel line markings, we use different numbers of hash marks when we have several groups of
equal sides. Draw a rectangle on the board and show the markings for equal opposite sides.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-17
Exercises: Mark the equal sides on the quadrilaterals in the previous exercises.
Answers:
a)
b)
c)
d)
Review special quadrilaterals. Draw on the board:
Have students copy the shape, then measure the sides to the closest millimeter, and label equal
sides with hash marks. ASK: How many equal sides does this shape have? (4) ASK: Do the
opposite sides in the picture look like they are parallel? (yes) SAY: In fact, the opposite sides
in this shape are parallel. This is a parallelogram. ASK: What do we call parallelograms with
four equal sides? (rhombuses) What other special quadrilaterals do you know? (rectangles,
squares, kites) What do you check to see if a quadrilateral is a rectangle? (if it has 4 right angles)
What do you check to see if a quadrilateral is a square? (if it has 4 right angles and 4 equal sides)
Do rectangles have parallel sides? (yes, opposite sides in a rectangle are parallel) Are rectangles
also parallelograms? (yes) Do rectangles have any pairs of equal sides? (yes, opposite sides in
a rectangle are equal)
To remind students of the relationships between special quadrilaterals, draw the Venn diagram
below on the board, explaining that if you had a bunch of quadrilaterals, this diagram would
allow you to sort them.
Quadrilaterals
Parallelograms
Rhombuses
Trapezoids
Rectangles
Point out that all rhombuses and all rectangles are also placed inside the group for
parallelograms. ASK: Where would all squares go on this diagram? (in the region that is
common to rhombuses and rectangles) Why are squares also rhombuses and rectangles?
D-18
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
(a square is a rhombus because it has 4 equal sides, and it is also a rectangle because it has
4 right angles) Have students copy the diagram so it fills a whole notebook page and draw a
shape in each region of the diagram, marking the parallel sides, right angles, and equal sides.
You might wish to distribute BLM Geometric Terms (2) as a summary of terms.
Activities 1–3
Use The Geometer’s Sketchpad® for the following three activities.
1. Constructing perpendicular and parallel lines.
a) Draw a line. Label it m.
b) Mark a point not on the line m. Label the point A.
c) Select the line and the point. Use the Construct menu options to construct a line n
perpendicular to m through point A.
d) Select line m and point A. Use the Construct menu options to construct a line p parallel to m
through point A.
e) Move or turn the line m. Does line n stay perpendicular to line m? Does line p stay parallel to
line m?
Answer: e) yes
2. Constructing a parallelogram.
a) Draw a line segment AB.
b) Draw a line segment BC, so that point C is not on line AB.
c) Using the Construct menu options, construct a line parallel to AB through C. Then construct a
line parallel to BC through A.
d) Mark and label D the intersection point of the lines you drew in part c).
e) Create a quadrilateral ABCD. Remember to use the option that shows the sides of a
quadrilateral. What type of quadrilateral is ABCD?
f) Measure the lengths of the opposite sides of ABCD. What do you notice?
g) Move the vertices of ABCD around. Does the quadrilateral change? Does it stay the same
type of quadrilateral? Does your observation in part f) hold when you move the vertices?
Answers: e) parallelogram; f) the opposite sides are equal, AB = CD, AD = BC;
g) the quadrilateral changes, but it stays a parallelogram and the opposite sides remain equal
3. Constructing a rectangle.
a) Draw a line segment AB.
b) Using the Construct menu options, construct a line perpendicular to AB through A.
c) Construct a line perpendicular to AB through B. Mark a point C on it.
d) Construct a line perpendicular to BC through C.
e) Mark and label D the intersection point of the lines you drew in parts b) and d).
f) Create a quadrilateral ABCD. What type of quadrilateral is ABCD?
g) Measure the lengths of the opposite sides of ABCD. What do you notice?
h) Move the vertices of ABCD around. Does the quadrilateral change? Does it stay the same
type of quadrilateral? Does your observation in part f) hold when you move the vertices?
Answers: f) rectangle; g) the opposite sides are equal, AB = CD, AD = BC; h) the quadrilateral
changes, but it stays a rectangle and the opposite sides remain equal
(end of activities)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-19
Extensions
1. Explain that a plane is a flat surface with no thickness. It extends forever along its length and
width. Parallel lines in a plane will never meet no matter how far they are extended in either
direction. Have students find a pair of lines not in the same plane that never meet and do not
intersect, but are not parallel. To prompt students to see the answer, sketch a cube on the
board and suggest that students look at the edges of the cube.
Sample answer: The thickened edges of the cube above are an example of parts of lines that
are not in a single plane, are not parallel, and never intersect.
2. Draw …
a) a hexagon with three parallel sides
b) an octagon with four parallel sides
c) a heptagon (7-sided polygon) with three pairs of parallel sides
d) a heptagon with two sets of three parallel sides
e) a polygon with three sets of four parallel sides
f) a polygon with four sets of three parallel sides
Sample answers:
D-20
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
(MP.3) 3. Discuss with students whether there can be more than one line perpendicular to a
given line through a given point and whether such a perpendicular always exists. You can use
the diagrams below to help students visualize the answers:
Answer: Imagine a line rotating around point P, as illustrated in the first diagram. At one
moment, it is perpendicular to AB, and then, as you continue to rotate the line, it will stop being
perpendicular to AB. You can also look at a line that is perpendicular to AB and slowly move it
from one end to the other. At one moment, it will touch point P, becoming the perpendicular to
AB through point P. Since P has no real width, the line will then immediately leave P. So there
cannot be more than one such line.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-21
G8-4
Measuring and Drawing Angles
Pages 79–81
Standards: preparation for 8.G.A.1, 8.G.A.5
Goals:
Students will measure and construct angles.
Prior Knowledge Required:
Knows what an angle is
Can identify right angles
Can identify and name angles and polygons
Understands the concept of measurement
Vocabulary: acute, base line, degree, endpoint, intersect, intersection point, obtuse,
opposite angles, origin, parallelogram, perpendicular, protractor, reflexive angles, rotation, scale
Materials:
pair of scissors or board compass for demonstration
several pieces of transparencies with angles drawn on them
overhead projector and markers
protractors
rulers
The Geometer’s Sketchpad®
BLM Using Protractors (Summary) (p. D-115)
geoboards and elastics (see Extension 1)
Review that the size of an angle is the amount of rotation between the arms. Remind
students that, in mathematics, the size of an angle is how much you need to turn one arm
extending from the vertex to get to the other arm extending from the vertex. In other words, the
size of an angle is the measurement of the rotation between the angle’s arms when the arms
are rays. Show students a pair of scissors or a board compass. Point out that the blades of the
scissors rotate around a peg, which is like a vertex. Hold the scissors so that one blade is
horizontal at all times. Open the scissors a little bit and then open them more and more to show
how the angle increases as the top blade rotates away from the horizontal blade. Trace a small
angle on the board and open the scissors wider than that angle; then show how the smaller
angle on the board “fits inside” the larger angle of the open scissors. Emphasize that the more
you rotate the blades, the more you open the scissors, and the wider the angle becomes.
You can also demonstrate the rotation using your arms; stand so that students look at your side
while you stretch your arms forward, one arm horizontal and the other raised somewhat. The
line of your shoulders acts as a vertex, and your arms act as the two arms of an angle. If you
keep one arm horizontal and swing the other up or down, you widen the angle between the
arms. The more you swing the one arm, the greater the angle.
D-22
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Draw on the board:
SAY: When the angle is small, the arms are only open a little bit. When the angle is large, the
arms are open a lot. Explain that we draw an arc to show how much the arms open or how
much we turn one arm to get to the other.
Size of the angle does not depend on the length of the arms in the picture of the angle.
Students often mistake the area between the arms of an angle for the size of the angle. To help
students who have this misconception, remind them that the angle is how much we need to turn
from one arm of the angle to the other. For any of the following activities that compares a pair of
angles, have students place a pencil on one of the arms of the angle and rotate it to get to the
other arm. Repeat with the second angle. The angle that needs more rotation is the larger
angle. For the size of the angle, it does not matter whether the arms of the angle in the picture
are short or long; the arms are rays and they can be extended as much as we need.
Draw the two angles below on two separate pieces of transparency and project them on the board:
NOTE: If an overhead projector is unavailable, you can do the following explanations by
drawing the angles on the board. You will have to skip the direct comparison and rely on
students estimating the size of the angles by eye.
Explain that although these angles look different and one picture looks larger than the other, the
angles are in fact the same size. In both these angles, you need to turn one arm the same
amount of rotation to get to the other arm. Slide one of the transparencies on top of the other to
show that the angles are the same. Point out that, just as we do not change a ray by extending
it, we do not change an angle by extending its arms. Extend the arms on the smaller picture to
show that the angles are exactly the same.
Draw the two angles below on two separate pieces of transparency and project them on the board:
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-23
ASK: Which angle is larger? Which angle requires you to rotate, or turn, more from one arm to
the other? (the one on the left)
Remind students that, even though the arc in the angle on the right looks larger, this does not
mean that the angle is larger; the sides of the angle are rays so they can be extended, and that
does not change the angle. Again, slide the transparencies one on top of the other to compare
the angles. Have a volunteer extend the arms of the angles. Students should clearly see that
the angle on the left is larger. Repeat with the pair of angles shown below:
Draw the pairs of angles in the exercises below, one pair at a time, on the board. For each pair,
have students decide which angle is larger, the angle on the left or the angle on the right. They
can signal the answers for each pair by pointing their thumbs to the left or right.
Exercises: Which angle is larger?
a)
b)
c)
Answers: a) left, b) left, c) right
Review degrees. Draw on the board:
SAY: These angles look to be about the same size. ASK: How can I check which angle is
larger? (measure them) In what units do we measure angles? (degrees)
Remind students that an angle that measures 1 degree is a very small angle. Have students
look at the picture in the top box on AP Book 8.1 p. 79 to see an angle that measures 1 degree.
Explain that we divide a full turn into 360 equal divisions and call them degrees.
A right angle measures 90 degrees. Draw on the board:
D-24
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Explain that this arc shows a full turn or full rotation around the intersection point of the lines.
ASK: What do you know about the lines in this picture? (they are perpendicular) How many right
angles does the picture show? (4) SAY: There are four right angles in a full turn or rotation, so
each right angle is a quarter turn. When you have a right angle with an arm that points straight
up (trace it on the picture) and an arm that points directly to the right (trace it on the picture), you
need to turn one arm a quarter turn to get to the other arm. ASK: If a full turn is 360 degrees,
how many degrees are in a quarter turn? (90 degrees) How do you know? (360 ÷ 4 = 90)
Introduce the notation for degrees. Explain that writing the word “degrees” takes time, so
people often use a symbol instead. The symbol is a small raised circle that is written after the
number. Write “90°” on the board as you SAY: for example, the measure of a right angle is
written as 90°.
Introduce acute and obtuse angles. Explain that angles that are smaller than a right angle are
called acute angles, and angles that are larger than a right angle are called obtuse angles. You
might point out the connections to these words: acute means “sharp” and obtuse means “blunt”
or “not pointed.”
Before assigning the following exercises, point out that even though a 1 angle is very small,
some angles can have a measure between two whole degree measures, so we can use
decimals or fractions to show degree measures. Decimals are more common.
Have students signal their answers in the next two exercises by making the letters A or O with
their hands.
Exercises:
1. Is the angle with this measurement acute or obtuse?
a) 35
b) 95
c) 78
d) 129
e) 90.25
Answers: a) acute, b) obtuse, c) acute, d) obtuse, e) obtuse, f) acute
2. Is the angle acute or obtuse?
a)
b)
c)
d)
f) 88.9
e)
Answers: a) acute, b) obtuse, c) obtuse, d) acute, e) obtuse
For each angle in Exercise 2 above, have students also say whether they expect the measure
of each angle to be more than 90° or less than 90°. (less for acute, more for obtuse)
Review protractors. ASK: What do we use for measuring angles? (a protractor) Have students
examine their protractors and say how they are similar to rulers and how they are different.
Draw attention to the fact that a protractor has two scales. Explain that having two identical
scales going in different directions allows you to measure the angles from both sides, but this
also means that you need to decide which scale you will use each time.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-25
Draw the picture on the left below on the board and explain that this protractor is like the
protractor in Question 3 on AP Book 8.1 p. 79. Explain that a good way to decide which scale
to use is to look at which scale starts with a zero on one of the arms of the angle. ASK: Which
scale has a zero on one of the angle arms, the inner scale or the outer scale? (the outer scale)
Cover or cross out the inner scale. ASK: What is the measure of the angle? (150) Is this angle
an acute angle or an obtuse angle? (obtuse) How do you know? (it is more than 90) Repeat
with the picture below on the right.
Have students complete Questions 3 and 4 on AP Book 8.1 pp. 79–80.
Placing protractors on angles. Point out the base line and the origin on a protractor (see gray
box on AP Book 8.1 p. 80). Have students find the base line and the origin on their own
protractors. Draw an angle on the board and demonstrate how to place a protractor correctly so
that the base line lines up with one arm of the angle and the origin is at the vertex. Point out that
this is similar to placing a ruler with the 0 at the beginning of the object you are measuring.
Have students use rulers to draw an acute angle in their notebooks and ask them to place their
protractors correctly. Circulate in the classroom to check that all students have done so. Then
have students measure the angle they drew. Repeat with an obtuse angle. Have students
exchange notebooks with a partner and measure each other’s angles to check their work.
Angles in a polygon. Draw a parallelogram on the board. Explain that we can measure angles
inside shapes. SAY: The measure of an angle inside the shape is the amount of rotation
between one arm (one side of the shape) and another arm (a different side of the shape). Draw
an arc to emphasize the angle between the sides, as shown below:
Point to the parallelogram on the board as you remind students that parallelograms have four
sides with two pairs of parallel sides and pairs of opposite sides that are equal in length. Have
students each a use ruler to draw a large parallelogram that is not a rectangle in their
notebooks, measure all four angles in the parallelogram, then exchange notebooks with a
partner and check their partner’s answers.
Opposite angles in a parallelogram are equal. Discuss students’ findings from measuring the
angles in the parallelogram above. ASK: Did you find any angles that had equal measurements?
(yes) Which angles were equal? (angles opposite each other in the shape, or opposite angles)
D-26
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Did everyone have the same parallelogram? (no) Did everyone get equal opposite angles?
(yes) Will all parallelograms have equal opposite angles? (yes)
Point out that the sides of a shape are often too short to conveniently measure the angle.
However, the sides of the shape are actually the arms of an angle starting at the vertex and
extending as rays, so you can extend these rays farther. Extend two sides of the parallelogram
you drew on the board. Point out that the parallelogram is inside the angle—it is part of the
space between the arms. Have a volunteer measure the angle.
Drawing angles. Model drawing an angle of 60 step by step, emphasizing the correct
positioning of the protractor:
Step 1: Draw a ray.
Step 2: Place the protractor on the ray. Line up the base line of the protractor on the ray. Line
up the origin of the protractor on the endpoint of the ray.
Step 3: Follow the scale that has a zero on the ray. Find the mark for 60. Make a mark beside
the protractor at the 60.
Step 4: Remove the protractor. Use a ruler to draw a ray from the endpoint to the mark.
Exercises:
1. a) Draw a line segment 7 cm long. Label it AB.
b) Use point A as a vertex. Draw an angle of 40. Extend the second arm of the angle so that it
is at least 12 cm long.
c) Use point B as a vertex. Draw an angle of 105 so that the second arm of the angle intersects
the ray you drew in part b).
d) Label the intersection point of the rays C. Measure the angle ACB.
Answer: d) ACB = 35
2. a) Draw a line segment 6 cm long. Label it DE.
b) Use point D as a vertex. Draw an angle of 53. Extend the second arm of the angle and mark
point F on it so that DF = 10 cm.
c) Draw the line segment EF.
d) Measure the sides and the angles of triangle DEF.
Answer: d) DEF = 90, DFE = 37, EF = 8 cm
Have students exchange notebooks and measure the sides and the angles of the triangles their
partners drew to check each other’s answers.
Constructing a perpendicular through a point. Draw a line on the board and mark a point P
outside it. Model the steps for constructing a perpendicular through a point that is not on the line:
Step 1: Place the protractor so that the line passes through the origin and the mark for 90.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-27
Step 2: Slide the protractor so that the flat side touches the point P.
Step 3: Draw a new line along the flat side of the protractor through point P and across the
original line.
Circulate among students to ensure that they are using the protractors and rulers correctly as
they work on the exercises below.
Exercises: Draw a pair of perpendicular lines that are on a slant—in other words, they are
neither vertical nor horizontal—and a point not on the lines. Draw perpendiculars to the slant
lines through the point. What quadrilateral have you constructed?
Answer: rectangle
Bonus: Draw a slant line and a point not on the line. Using a protractor and a ruler, draw a
square that has one side on the slant line and one of the vertices at the point you drew.
Answer: Draw a perpendicular through the given point to the given line. Measure the distance
from the point along the perpendicular to the line. Then mark a new point on the given line that
is that same distance from the intersection as the given point. Draw a perpendicular to the given
line through this point as well. Finally, draw a perpendicular to the last line through the given
point. Alternatively, when drawing the second last perpendicular, mark a point on that
perpendicular that is the same distance from the given line as the given point is, and on the
same side of the given line as the given point. Join this new point to the given point.
In Activity 1, below, students will draw and measure angles to reinforce the understanding that
the size of the angle does not depend on the length of the arms. They will learn to draw angles
of a given measure and understand the need to follow the specific procedures for doing so. In
Activity 2, students will practice measuring angles of polygons. In Activity 3, students will learn
to construct right triangles, including the need to use specific tools. In Activity 4, students will
construct a parallelogram and review the fact that its opposite angles are equal.
Activities 1–4
Use The Geometer’s Sketchpad® for the following four activities.
(MP.5) 1. Drawing and measuring angles.
a) Teach students to draw and measure angles by following these steps:
Step 1: Draw ray AB.
Step 2: Draw ray AC.
Step 3: Mark and label point D on AB and point E on AC.
Step 4: Select points in this order: D, A, E. Use the Measure menu options to measure angle DAE.
Have students move points D and E. Draw students’ attention to the fact that the points stay on
the rays they were drawn on and that the angle measure does not change. Have students move
D-28
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
points A, B, or C. ASK: Does the angle measure change? (yes, because the placement of the
rays change) Do points D and E stay on the rays? (yes) Students will need this angle for
parts b) and d), below.
b) Have students modify the angle they drew in part a) to try to make the angle measure
exactly 30. ASK: Is this hard or easy to do? (hard) Have students move the angle around to
see that the measure changes again.
c) Teach students to draw an angle of 30 by following these steps:
Step 1: Draw ray FG.
Step 2: Select vertex F. Use the Transform menu options (Mark center) to mark F as the center
of rotation.
Step 3: Select the ray. Use the Transform menu options to rotate the ray around the marked
center. When selecting the angle of rotation, remember to mark it as degrees.
Have students measure the angle. They will need to create points on the arms. Have students
move the angle and the points on the arms to see how it changes. ASK: Does the angle
measure change? (no)
d) Teach students to draw an angle equal to DAE from part a) by following these steps:
Step 1: Draw and label point J away from angle DAE. Construct a ray JK.
Step 2: Mark point J as a center of rotation.
Step 3: Select the measure of DAE. Use the Transform menu options (Mark angle) to mark
this measure as the angle of rotation.
Step 4: Select ray JK. Use the Transform menu options (Rotate) to rotate the ray around J by
the selected angle. Mark a point on the new ray and label it M.
Have students measure angle KJM. Have them try to modify both angles DAE and KJM to see
that the angle measurements stay equal even when angle DAE changes.
(MP.5) 2. Measuring angles of polygons. Have students draw polygons and measure the
sizes of the angles and the lengths of the sides of these polygons. Have students check that the
angle measurements they find make sense. For example, the software sometimes measures
angles in the wrong direction, producing an answer less than 180° for reflexive angles
(e.g., angles such as ABC in the quadrilateral below).
D
A
B
C
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-29
(MP.5) 3. Constructing right triangles. Have students draw a triangle using the Polygon tool.
Ask them to move the points around to make it look like a triangle with a right angle. Remind
students this is called a right triangle. Then ask them to measure the angles of the triangle and
to check whether it is indeed a right triangle. ASK: Is it easy to draw a perfect right triangle this
way? (no) If you move the points around, does the triangle remain a right triangle? (no) Have
students think about how they draw a right triangle on paper. ASK: What instruments do you
use and why? (a protractor, to make sure the triangle has a right angle) What could we use
instead of protractors in The Geometer’s Sketchpad®? (perpendicular lines) Have students
draw a right triangle in The Geometer’s Sketchpad® by following these steps:
Step 1: Draw a line segment AB.
Step 2: Draw a line perpendicular to AB through point B.
Step 3: Mark a point anywhere on the perpendicular line you drew and label it C.
Step 4: Use the Polygon tool to construct a triangle ABC.
Have students check that triangle ABC remains a right triangle even if the vertices are
moved around.
4. Angles in a parallelogram.
a) Draw and label a line segment AB. From B, draw another line segment so that they form an
angle. Label the second line segment BC.
b) Draw a line parallel to AB through C.
c) Draw a line parallel to BC through A.
d) Mark the intersection point of the lines you drew in parts b) and c). Label it D.
e) Use the Polygon tool to create a quadrilateral ACBD. What type of quadrilateral is ACBD?
f) Measure the angles in ACBD. What do you notice about the angles that are opposite?
g) Move the vertices of ACBD. Does the type of quadrilateral change? Do the angles change?
Do the opposite angles stay equal?
Answers: e) parallelogram, f) opposite angles are equal, g) the type of quadrilateral does not
change, the angles change, the opposite angles stay equal
(end of activities)
For a summary related to angles and using protractors, you can provide students with
BLM Using Protractors (Summary).
Extensions
1. Students can use geoboards and elastics to make right, acute, and obtuse angles. When
students are comfortable doing that, they can create figures with the following given angles.
a) a triangle with 3 acute angles
b) a quadrilateral with 0, 2, or 4 right angles
c) a quadrilateral with 1 right angle
d) a shape with 3 right angles
e) a quadrilateral with 3 acute angles
D-30
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Sample Answers:
Students can also try to create polygons similar to those created on geoboards using protractors
and rulers.
(MP.4) 2. a) What is the angle between the hands of an analog clock at 3:00? Can you tell
without using a protractor?
b) A minute hand rotates from one mark to the next, every minute. What is its angle of rotation
in 1 minute?
c) How many degrees does the hour hand turn in 1 hour? In 1 minute?
d) The time is 12:24. How much did the hour hand turn from the vertical position (12)?
e) The time is 12:24. How much did the minute hand turn from the vertical position (12)?
f) What is the angle between the hands on an analog clock at 12:24?
g) Find the angle between the hands on an analog clock at 1:36 and again at 3:48.
Answers:
a) At 3:00 the minute hand points to the 12, so straight vertical. The hour hand points to the 3,
so straight horizontal. The angle is 90.
b) An hour is 60 minutes and a whole circle is 360. Each minute the minute hand turns
360 ÷ 60 = 6.
c) An hour hand makes a full 360 turn in 12 hours, so it turns 360 ÷ 12 = 30 every hour.
Each hour is 60 minutes, so the hour hand turns 30 ÷ 60 = 0.5 every minute.
d) The hour hand turned 24 × 0.5 = 12.
e) The minute hand turned 24 × 6 = 144.
f) The angle between the hands at 12:24 is 144 − 12 = 132.
g) The angle between the hands at 1:36 is 216 − 48 = 168 (48 for the hour hand because
30 for 1 hour + 36 × 0.5 for minutes = 18), and the angle between the hands at 3:48 is
288 − 114 = 174.
3. Use The Geometer’s Sketchpad® to create:
a) a triangle with each side equal to 3 cm and each angle equal to 60
b) a quadrilateral with each side equal to 3 cm and each angle equal to 90
c) a pentagon with each side equal to 3 cm and each angle equal to 108
d) a hexagon with each side equal to 3 cm and each angle equal to 120
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-31
G8-5
Sum of the Angles in a Triangle
Pages 82–83
Standards: 8.G.A.5
Goals:
Students will informally establish that the angles in a triangle add to 180° and use this to find the
angles in a triangle.
Prior Knowledge Required:
Can identify and name angles and polygons
Can measure and construct angles
Knows that angle measures are additive
Vocabulary: endpoint, intersect, intersection point, reflexive angles, straight angle
Materials:
BLM Using Protractors (Summary) (p. D-115, optional)
protractors
The Geometer’s Sketchpad® (optional)
BLM Sum of the Angles in a Triangle and a Quadrilateral (p. D-116)
BLM Sum of the Angles in a Polygon (p. D-117, see Extension 1)
NOTE: To have students discover the fact that interior angles of a triangle add to 180, you can
either work through the introduction below or do Activity 1.
Discovering the sum of the angles in a triangle. Draw an angle on the board. Remind
students how to place a protractor correctly to measure an angle. Have a volunteer measure the
angle on the board. Remind students how to construct an angle of a given measure. For example,
you might ask students to draw two angles, 35 and 123, exchange notebooks with a partner,
and check each other’s work. To review drawing angles, students could refer to BLM Using
Protractors (Summary).
Have students construct a triangle with angles 50 and 30, and measure the third angle in the
triangle. (100) Repeat with a triangle with angles 90 and 20. (70) Have students add the
angles in both triangles. ASK: What do you notice? (the three angles always add to 180)
Ask students to construct a triangle of their choice, measure the angles, and add them. Have
partners to exchange notebooks and check each other’s work. ASK: Did everyone get the same
triangle? (no) Did everyone get angles adding to 180? (yes) If anyone answers no, have them
check the measurements to find the mistake. SAY: Angles in a triangle always add to 180.
D-32
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Activity 1
Use The Geometer’s Sketchpad® for this activity.
Sum of the angles in a triangle. Have students construct a triangle using the Polygon tool.
Then have them measure the angles of the triangle. Remind students to check that each angle
measure makes sense: they need to be smaller than 180.
Show students how to add the angles in the triangle using the Number menu option: Use the
Calculate option in the Number menu options. In the calculation window, write an expression to
add the angle measures. Click on each angle measure to make them appear in the expression.
Add the three angle measures. Have students move the vertices of the triangle around and
watch how the angles change, but the sum remains 180.
(MP.5) Point out that, when we use protractors, we cannot measure angles with great precision.
Software measures angles with better precision, but it displays the rounded answer. If you add
the answers by hand, you might not get 180. The software adds the angles before rounding them.
(end of activity)
Finding the measure of the angles using the sum of the angles in a triangle. Draw a
triangle on the board and write the measure of two of the angles in the triangle. ASK: How can I
find the measure of the third angle? (subtract: 180 minus the sum of the other two angles)
Work through the first exercise below together and then have students work individually.
Exercises: Find the missing angle in the triangle.
a)
b)
c)
A
D
45
53
C 64 65 B
E
47
d) S
H
F
G
45
Answers: a) 51, b) 80, c) 90, d) 44
K
115
U
21
N
(MP.2) Finding the measure of the angles using information presented symbolically.
Explain that we can usually use an equal number of small arcs to show that the angles in a
diagram are equal. For example, we could change the diagram in part c) in the exercises above
by erasing the measure of one of the angles, G or H, and marking the two angles with an arc.
Show the change on the diagram, as shown below:
H
45
K
G
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-33
Draw on the board:
M
76
L
K
ASK: What do the angles in this triangle add to? (180) What is the sum of angles K and L?
(104) How do you know? (180 − 76 = 104) What does the diagram tell us about the sizes of
angles K and L? (they are the same) What is the size of each of these two angles? (52) How
do you know? (104 ÷ 2 = 52)
Before assigning the next exercises, remind students that right angles are labeled with a small
square and that a right angle measures 90.
Exercises: Find the missing angle in the triangle.
a)
b)
A
D
E
C
63
c)
A
32
F
55
C
N
B
Answers: a) 27, b) 58, c) 70
SAY: Suppose a triangle has two equal angles. One of the angles in this triangle is 90. What
are the sizes of the other two angles? ASK: Can the equal angles be 90 each? (no) Why not?
(the equal angles would add to 180, leaving no room for the third angle) Invite a volunteer to
draw the triangle on the board and mark the measures of the angles. Then have students find
the size of the two missing angles in the triangle. (45)
(MP.2, MP.3) Present a similar problem. SAY: A triangle has two equal angles. One of the
angles in this triangle is 50. ASK: What are the sizes of the other two angles? Give students
several minutes to think and then ASK: How is this problem different from the previous
problem? (the given angle is an acute angle, not a right angle) Can a triangle have two angles
of 50? (yes) What is the third angle then? (80) Sketch the triangle on the board and ask
volunteers to mark the angles on the picture. Then draw on the board:
50
ASK: Can this situation happen? (yes) What are the measures of the other two angles? (65)
Sum of the angles in a quadrilateral. Ask students to draw a quadrilateral, paying no attention
to side lengths. Ask them to measure the angles in the quadrilateral and add them together.
Have students exchange notebooks and check each other’s answers. Discuss the results:
students should see that the angles in each quadrilateral add to 360 even though the
D-34
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
quadrilaterals are different. You can repeat Activity 1 above using The Geometer’s Sketchpad®,
but drawing quadrilaterals instead of triangles as an alternative to measuring with a protractor.
Before having students work with the software, draw on the board:
D
B
A
C
Explain that angles such as angle B are called reflexive angles. Point out that, with angle B, to
turn from one arm to the other arm you need to make more than half a turn, so B measures
more than 180. Point out to students that, if they create a quadrilateral with one reflexive angle,
the software measures the angle incorrectly, and the sum of the angles changes, producing an
incorrect result. They should avoid modifying quadrilaterals so that reflexive angles appear.
You can also work through the informal proof of the sum of the angles in triangles and
quadrilaterals in Activity 2.
Activity 2
Remind students that an angle that is made by two rays going in opposite directions creating a line
is called a straight angle and measures 180. Remind them also that a full turn measures 360.
Draw on the board:
ASK: What is the sum of the angles in this picture? (360) As a prompt, place a pencil along one
of the arms and SAY: If an arm rotates through all of these angles, it will return back to this
position, making a full turn. Turn the pencil to illustrate the rotation.
Give students BLM Sum of the Angles in a Triangle and a Quadrilateral and have them cut
out the triangle. Have them cut off the angles along the lines marked on the triangle and place
them together to create a straight angle. Then have them answer the questions. (1. b) yes,
c) 180, 180) Repeat with the quadrilateral, this time showing that the angles together make
360. Then have them answer the questions. (2. c) 360, 360)
(end of activity)
(MP.3) Proving that sum of the angles in a quadrilateral is 360 using logic. SAY: I would
like to use logic to explain why the sum of the angles in a quadrilateral is 360. Draw on
the board:
A
D
B
C
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-35
Ask students to think of how they could explain why the sum of the angles in ABCD is 360.
Have them discuss their explanations in pairs and then in groups of four. Debrief as a class.
The discussion should include the following:
The sum of the angles in each triangle is 180.
Some angles in the triangles are the same angles as in the quadrilateral (B and D);
other angles combine to become angles in the quadrilateral:
DAC + CAB = DAB
DCA + ACB = DCB
Thus the sum of the angles in the quadrilateral is the same as the sum of the angles in
the two triangles combined.
Extensions
(MP.7) 1. Have students work through BLM Sum of the Angles in a Polygon.
Selected Answers:
b)
Number of
Expression for
Sum of
Number
Triangles
the Sum of
Interior
Polygon
of
Created by
Interior Angles
Angles
Sides
Diagonals
Quadrilateral
4
2
180 × 2
360
Pentagon
5
3
180 × 3
540
Hexagon
6
4
180 × 4
720
Heptagon
7
5
180 × 5
900
Octagon
8
6
180 × 6
1080
Nonagon
9
7
180 × 7
1260
Decagon
10
8
180 × 8
1440
n-sided polygon
n
n−2
180 × (n − 2)
180 × (n − 2)
2. In the diagram below, triangle EFG is a right triangle and FGH = FHG. Also, EGF = 26
and EFH = 42. Find the measure of GHF.
E
42
F
H
26
G
Solution: From the sum of the angles in triangle EFG, EFG = 180 − (90 + 26) = 64.
Then HFG = 64 − 42 = 22. From the sum of the angles in triangle FGH,
FGH + FHG = 180 − 22 = 158. Since FGH = FHG, they both measure 158  2 = 79.
D-36
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-6
Triangles
Pages 84–85
Standards: 8.G.A.5
Goals:
Students will classify triangles by the number of equal sides and by the size of angles.
Students will use the properties of triangles to find the size of angles in the triangles.
Prior Knowledge Required:
Can identify and name angles
Can measure angles
Knows that angles in a triangle add to 180º
Vocabulary: acute-angled triangle, acute triangle, equilateral, intersect, intersection point,
isosceles, obtuse-angled triangle, obtuse triangle, right-angled triangle, right triangle,
scalene
Materials:
BLM Triangles for Folding (p. D-118)
scissors
blank paper cut into the shape of a hand-drawn circle
protractors
BLM Geometric Terms (pp. D-111–113, see Extension 2)
Introduce classification of triangles by angles. Draw a few different triangles on the board,
including acute, obtuse, and right triangles. Number the angles in each as 1, 2, and 3. Have
students identify the largest angle in each triangle by raising the number of fingers equal to the
number in the angle. Circle the label for the largest angle and ask students to identify the angle
as acute, obtuse, or a right angle.
Explain that triangles are classified by the size of the largest angle. Triangles in which the
largest angle is acute are called acute-angled or acute triangles; triangles in which the largest
angle is a right angle are called right-angled or right triangles; and triangles in which the largest
angle is obtuse are called obtuse-angled or obtuse triangles. Have students identify each
triangle on the board as acute, right, or obtuse.
NOTE: Students might be familiar with a different form of classification from earlier grades: a
triangle that has an obtuse angle is an obtuse triangle, and a triangle that has a right angle is a
right triangle. Triangles that have no obtuse and no right angles have three acute angles, and so
are called acute triangles. If the issue arises, discuss with students that the definitions are
essentially equivalent. ASK: Can a triangle can have more than one obtuse or more than one
right angle? (no) Why not? (the sum of the triangle’s three angles would be more than 180)
Point out that this means that the largest angle in a triangle that has an obtuse angle is the
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-37
obtuse angle, so the triangle will be called obtuse by both definitions. The same argument
applies to the right triangle. Finally, if the largest angle is acute, then all three angles in the
triangle are acute, and the triangle is an acute triangle by both definitions.
Classifying triangles by angles. Draw on the board:
Ask students to identify the largest angle. (angle 1) ASK: What type of triangle is this? Students
are likely to say this is a right triangle. Invite a volunteer to check using a protractor. The triangle
is an acute triangle because the largest angle is 85°. SAY: This is why it is important to always
check the measure of an angle that looks like a right angle.
Write on the board:
right triangle
acute triangle
obtuse triangle
Draw the triangles in the exercises below one at a time and have students signal the answer to
the exercises by pointing in the direction of the type of triangle (left for acute triangles, up for
right triangles, or right for obtuse triangles). After students signal the answer for part d), have a
volunteer check that the angle that looks like a right angle is indeed a right angle.
Exercises: Classify the triangle as acute, right, or obtuse.
a)
b)
c)
d)
Answers: a) acute, b) obtuse, c) acute, d) right
Using the sum of the angles in a triangle to classify triangles. Remind students that angles
in a triangle add to 180. Review finding the measure of the third angle from the measures of
the other two. Draw on the board:
C
A
41
44
B
Ask students to find the measure of angle C. (95) Have a volunteer explain how they found the
answer. (180 − (41 + 44) = 180 − 85 = 95) ASK: Which angle is the largest angle in the
D-38
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
triangle? (C) Is angle C a right, acute, or obtuse angle? (obtuse) What type of triangle is ABC?
(obtuse triangle)
ASK: How could you figure out that ABC is an obtuse triangle if I have only given you the
measures of angles A and B, without drawing the triangle? (use the same method: find the
measure of the third angle, decide which angle is the largest, and then check what type of angle
it is) SAY: Triangle DEF has one angle of 35 and another angle of 45. ASK: What is the
measure of the third angle? (100) How do you know? (180 − (35 + 45) = 100) Have a
volunteer write the calculation on the board. ASK: What is the largest angle in the triangle, 35,
45, or 100? (100) What type of triangle is DEF? (obtuse triangle)
Exercises: What type of triangle is triangle GHI?
a) G = H = 47
b) G = 50, I = 47
c) I = 63, H = 27
d) I = H = 59
e) G = H = 36.5
f) G = 90, I = 35
Selected solution: a) I = 180 − (47 + 47) = 86, so the largest angle is I and it is an acute
angle, so triangle GHI is an acute triangle
Answers: b) acute, c) right, d) acute, e) obtuse, f) right
(MP.8) SAY: In one of the questions in the previous exercises you could find the answer without
finding the third angle. ASK: Which question was that? (part f) How could you solve this problem
without finding the size of the third angle? (G is 90, a right angle, so it has to be the largest
angle in the triangle, and thus the triangle is a right triangle)
Introduce classification by triangle side lengths. Explain that another way to classify
triangles is by using side lengths. Remind students that we mark sides of equal length with
equal numbers of hash marks. SAY: Triangles with at least two equal sides are called isosceles
triangles. Triangles that have no equal sides are called scalene triangles.
Exercises: Classify the triangles as isosceles or scalene.
a)
b)
c)
d)
e)
Answers: a) isosceles, b) scalene, c) isosceles, d) scalene, e) isosceles
Introduce equilateral triangles. Explain that shapes with all sides equal are called equilateral.
For example, a rhombus is an equilateral quadrilateral. Triangles with three equal sides are
called equilateral triangles. ASK: Are equilateral triangles isosceles or scalene triangles?
(isosceles triangles) Is one of the triangles in the previous exercises equilateral? (yes)
Which one? (part e))
Classify triangles using both classifications. Explain that you can label triangles using both
classifications, according to the largest angle and by the number of equal sides. For example, if
you fold a rectangular sheet of paper along the diagonal, you get a right scalene triangle.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-39
Exercises:
1. Classify the triangles in the previous exercises using both classifications.
Answers: a) acute isosceles triangle, b) obtuse scalene triangle, c) acute isosceles triangle,
d) right scalene triangle, e) acute equilateral triangle
2. On grid paper draw:
a) a right isosceles triangle
c) an obtuse isosceles triangle
Sample answers:
a)
b)
b) a right scalene triangle
d) an acute scalene triangle
c)
d)
Activity 1
Isosceles triangles also have equal angles. Remind students that they can use paper-folding
to check if sides or angles of triangles are equal. Give students BLM Triangles for Folding and
have them cut out the triangles. Have students fold the triangles so that they can see which are
isosceles and which are scalene. Point out that this folding also compares two of the angles in
the triangle. Have students label each triangle with its type and mark the equal angles with the
equal number of arcs. (isosceles triangles: A, D, F, H, I; scalene triangles: B, C, and E; triangle
H is an equilateral triangle; triangle G is a right scalene triangle)
ASK: What do you notice about the angles in isosceles triangles? (there are at least two equal
angles) Are there equal angles in scalene triangles? (no)
(end of activity)
SAY: In an isosceles triangle, the angles between the equal sides and the third side are always
equal. Explain also that, if a triangle has two equal angles, it is always an isosceles triangle.
Draw on the board:
B
A
C
ASK: Which sides are equal in this triangle? (AB and BC) Point out that the equal angles are
both adjacent to the third, unequal side of the triangle.
Size of the angles in an equilateral triangle. ASK: Are there any equilateral triangles in your
collection from BLM Triangles for Folding? (yes, triangle H) What do you notice about its
angles? (they are all equal) How do you know? (when we fold the triangle to check the angles,
we find that any 2 angles are the same and the sides are the same length; the same is true
when we check the next side)
D-40
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
(MP.2) SAY: I would like to find the size of the angles in an equilateral triangle without actually
measuring them. ASK: How could we do it? Give students a few minutes to think and then
discuss the solution in pairs. PROMPT: What is the sum of the angles in any triangle? (180)
What do we know about the angles in an equilateral triangle? (they are all equal) What is the
size of one angle? (180 ÷ 3 = 60) Have students measure the angles in triangle H from BLM
Triangles for Folding to check.
Finding angles in isosceles triangles. Work through the first two problems in the following
exercises as a class. Then have students solve the rest of the exercises individually.
Exercises: Find the missing angles in the isosceles triangle.
a)
b)
c)
A
D
W
63
C
B
V
124
E
d)
24
F
U
R
60
O
N
Sample solution: a) A = C = 63, so B = 180 − (63 + 63) = 180 − 126 = 54
Answers: b) E = D = 28, c) U = 24, W = 132, d) R = N = 60
Activities 2–4
2. Creating right angles. Show students a piece of paper cut in the shape of a hand-drawn
circle. Ask them how they could make a right angle from this piece of paper. Have students
draw a circle by hand, cut it out, and use it to test their ideas.
To prompt students to see the solution, remind them that there are 360° in a full turn, and they
need 90°. ASK: What fraction of the whole turn is a right angle? (a quarter turn) How can you
fold the paper into four equal parts? (fold the shape in half, and then fold it a second time so that
the crease is folded against itself) Point out that students have made a right angle, which they
can use to distinguish between right, acute, and obtuse angles. All they need to do to classify
angles is compare the angle in question to the right angle they created. Students can use the
right angle they created as the square corner they use to distinguish between right, acute, and
obtuse angles in Question 2 on AP Book 8.1 p. 84.
3. Have students use the right angle made from the circle-shaped piece of paper from Activity 2
to create a square. If necessary, remind students that they can use the circle to make a right
angle and ask students about the defining properties of squares.
(MP.2) 4. Have students create a square from a large rectangular sheet of paper following the
steps below:
Step 1: Fold the short side of the paper down onto the long side to create a right isosceles
triangle. The extra part will be a rectangle.
Step 2: Fold the extra part of the page—a rectangle—over the triangle.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-41
Step 3: Unfold the paper and cut off the rectangle. (see diagram below)
Now have students create a special triangle. Fold the square in half vertically (not diagonally) so
that a crease divides the square into two rectangles. (see a), below) Fold the top right corner of
the square down so that the top right vertex (or corner) touches the crease. The vertex should
be slightly above the bottom edge. (see b), below) Mark the point where the corner touches
crease and trace a line along the folded-over edge of the square with a pencil. (see c), below)
Unfold. (see d), below) With the top left corner of the square, repeat steps b) to d). The top left
corner will touch the crease at the same point, which is the vertex of your new triangle. (see e),
below) Cut the triangle out along the traced lines.
What type of triangle have you created? Explain.
Answer: I started with a square and ensured in step b) that the sides of the triangle are equal to
the side of the square. So it is an equilateral triangle.
(end of activities)
Extensions
(MP.1) 1. Remind students that the longest side in a triangle is always shorter than the sum of
the other two sides. For example, you cannot make a triangle with sides 1 cm, 2 cm, and 3 cm.
How many triangles with sides that are a whole number of centimeters in length are there for the
perimeter? For each triangle, say what type of triangle it is.
a) perimeter = 3 cm
b) perimeter = 4 cm
c) perimeter = 5 cm
d) perimeter = 7 cm
e) perimeter = 8 cm
f) perimeter = 9 cm
Answers: a) 1 equilateral triangle with all sides = 1 cm; b) no triangles; c) 1 isosceles triangle
with sides of 2, 2, and 1 cm; d) 2 isosceles triangles with sides of 3, 2, and 2 cm or 3, 3,
and 1 cm; e) 1 isosceles triangle with sides of 3, 3, and 2 cm; f) 3 triangles: an equilateral
triangle with all sides = 3 cm, 1 isosceles triangle with sides of 4, 4, and 1 cm, 1 scalene triangle
with sides of 2, 3, and 4 cm
(MP.7) 2. Remind students that a Venn diagram can group shapes that have similar and
different properties—for example, by showing shapes that have one property or the other
property, both properties, or neither property. Remind them that shapes that have neither
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
property are placed outside of both groups. Write these properties on the board for use on
Venn diagrams:
Group 1
Acute
Right
Obtuse
Group 2
Scalene
Isosceles
Equilateral
Ask students to pick a property from each column and make a Venn diagram about triangles
using both properties. Point out that, in some cases, a Venn diagram will have an empty region.
Also, remind them that equilateral triangles are also isosceles, so when they are sorting
triangles using “isosceles,” equilateral triangles should be within this group. Students can refer
to BLM Geometric Terms as needed.
When students finish, ask them to try to draw an example triangle in each region of their Venn
diagram. If they cannot manage to produce a triangle in one of the regions, ask them to explain
to a partner what problems they encountered and to think together whether a triangle in that
region is possible.
Selected answers:
Triangles
Right
Equilateral
There is no intersection between the two groups because equilateral triangles have all angles
equal to 60 and so cannot have a right angle. The triangle on the far right is an acute scalene
triangle, so is not a right triangle and not equilateral.
(MP.2) 3. Classify triangle ABC if АВ + ВС = АВ + АС = ВС + АС.
Solution: АВ + ВС = АВ + АС, so BC = AC, and АВ + АС = ВС + АС, so AB = BC,
and AB = BC = AC, and the triangle ABC is equilateral.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-43
G8-7
Making a Geometric Sketch
Pages 86–89
Standards: 8.G.A.5
Goals:
Students will make quick sketches for problems, identifying the relevant information and solving
problems using a sketch.
Prior Knowledge Required:
Can identify equal sides, equal angles, and right angles
Is familiar with standard markings for equal sides, equal angles, and right angles
Can classify triangles
Knows that isosceles triangles have equal angles adjacent to the unequal side
Can name angles and polygons and can identify a named angle or polygon
Knows that the sum of the angles in a triangle is 180 and the sum of the angles in a
quadrilateral is 360
Vocabulary: acute-angled triangle, acute triangle, endpoint, equilateral, intersect,
intersection point, isosceles, obtuse-angled triangle, obtuse triangle, right-angled triangle,
right triangle, sketch
Review. Remind students how to mark equal sides, equal angles, and right angles. Review with
students the classification of triangles according to the number of equal sides and the size of the
angles and the properties of special quadrilaterals related to sides and angles. You might wish
to draw the table below on the board, which summarizes this classification:
Quadrilateral
Properties
parallelogram
2 pairs of equal parallel sides, equal opposite angles
rectangle
parallelogram, 4 right angles
rhombus
parallelogram, 4 equal sides
square
rectangle, rhombus, 4 right angles, 4 equal sides
trapezoid
exactly 1 pair of parallel sides
Remind students that the angles of an equilateral triangle are equal to 60, and the angles that
equal sides of an isosceles triangle make with the third side are equal.
Making a sketch. Explain that a sketch is a quick drawing made without using instruments such
as a ruler or protractor. SAY: Knowing how to make a sketch is an important math skill.
Sketches can help us organize information, visualize described objects, see relationships,
explore ideas, and solve problems.
D-44
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Have students complete Questions 1–15 on AP Book 8.1 pp. 86–89. You can use the
exercises below as extra practice for students who work more quickly than others or for
students who are struggling with a particular concept or step.
Drawing diagrams accurately.
Exercises:
1. Sketch an obtuse isosceles triangle and an acute isosceles triangle.
Sample Answers:
2. Which sketch is better, A or B?
A.
B.
A.
A.
A.
B.
B.
B.
Answers: a) B, b) A, c) B, d) A
3. Sketch the figures.
a) line segment AC with AB = BC
b) a rectangle 2 cm by 6 cm
c) a rhombus with angles 20° and 160°
d) triangle KLM with sides 5 cm, 3 cm, and 3 cm
Sample Answers:
6 cm
a)
b)
c)
A
B
C
2 cm
3 cm
d)
20
160
L
K
5 cm
M
Drawing the shapes as generally as possible and not adding extra information
accidentally. Explain that we need to draw shapes matching the description as precisely as we
can, but it is important not to add something that is not given. For example, if we know a shape
is a rectangle, we should not sketch a square. Though the square is a rectangle, squares have
equal adjacent sides, not just equal opposite sides.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-45
Exercises:
(MP.6) Which sketch adds more information than in the description, A or B? Explain.
a) Parallelogram ABCD has diagonals AC and BD.
A. B
B.
B
C
A
C
D
A
D
b) Quadrilateral EFGH has perpendicular diagonals EG and FH.
F
F
A.
B.
E
E
G
H
G
H
Answers: a) Sketch B gives too much information—it meets the criteria but shows a rhombus,
not a general parallelogram; b) Sketch A gives too much information—it meets the criteria but
seems to have equal sides, so it is a rhombus
Solving problems that require making different sketches.
Exercises:
1. A parallelogram is made from two isosceles triangles, each with two sides of 3 cm and
one side of 5 cm. What is the perimeter of the parallelogram? Make two sketches to show
the different placement of the triangles in the parallelogram. Then solve the problem.
Answer:
3 cm
3 cm
5 cm
Perimeter = 4 × 3 cm = 12 cm
5 cm
Perimeter = 2 × 3 cm + 2 × 5 cm = 16 cm
2. A parallelogram is made from two isosceles triangles that have an angle of 98 each. What
are the angles of the parallelogram? Can the angle of 98 be one of the angles between one of
the equal sides and the third side? Make two sketches to show the different placement of the
triangles in the parallelogram. Then solve the problem.
Answers: The 98 angle is an obtuse angle, so it has to be the angle between the two equal
sides of the isosceles triangle:
98
41 98
Angles: 98, 98, 82, 82
D-46
Angles: 41, 41, 139, 139
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Ignoring irrelevant information.
Exercises: Make a sketch for the problem (without solving it). Ignore the unnecessary information.
A traffic island has the shape of a parallelogram with one of the angles 65º. The island
contains three shrubs and a circular flowerbed 1 m wide. What are the sizes of the angles
of the traffic island?
Bonus: Solve the problem.
Answers:
65
Bonus: 65º, 115º, 65º, 115º
Adding information that can be deduced.
Exercises: Add to the sketch other information you can deduce. Then solve the problem.
In triangle ABC, AB = BC and ∠A = 50. Point D is on the line segment AC, so that BD  AC.
What is the size of CBD?
Answer:
B
A
50
50
D
C
CBD = 180 − (90 + 50) = 40
Solving problems by making sketches.
Exercises: a) The shorter side of a parallelogram is 5 cm. The longer side is 2 cm longer than
the shorter side. What is the perimeter of the parallelogram?
b) A square is cut into two identical parts and rearranged to make a rectangle. The short side of
the rectangle is 6 cm. How long is the long side of the rectangle?
c) A square is cut into two identical parts and rearranged to make a triangle. What are the
angles of the triangle?
Answers: a) 24 cm; b) 24 cm; c) 45, 45, 90
Extension
(MP.1) Make a sketch to find the answer.
Ron drew rhombus KLMN and its diagonal NL. He measured NML = 96.
a) What is the size of NKL?
b) What is the size of KNL?
Explain how you know.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-47
Answers:
K
N
L
96
96
M
a) Since KLMN is a rhombus, it is a parallelogram, its opposite angles are equal, and
NML = NKL = 96.
b) Since KLMN is a rhombus, KL = KN, so triangle KNL is isosceles, and NLK = KNL. From
the sum of the angles in a triangle, NLK + KNL = 180 − 96 = 84, so NLK = KNL = 42.
D-48
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-8
Counterexamples
Pages 90–92
Standards: preparation for 8.G.A.5, 8.G.B.7
Goals:
Students will recognize and create counterexamples.
Prior Knowledge Required:
Can identify polygons
Can identify vowels and consonants
Can identify even and odd numbers
Is familiar with standard markings for equal sides, equal angles, and right angles
Can classify triangles
Knows that isosceles triangles have equal angles adjacent to the unequal side
Can classify quadrilaterals
Knows the basic properties of special quadrilaterals
Vocabulary: acute triangle, counterexample, equilateral, false, isosceles, obtuse triangle,
right triangle, true
Materials:
Anno’s Hat Tricks by Akihiro Nozaki and Mitsumasa Anno (see Extension 2)
BLM Sudoku—Warm-Up (pp. D-119–120, see Extension 3)
BLM Sudoku—Introduction (pp. D-121–122, see Extension 3)
BLM Sudoku—Another Strategy (p. D-123, see Extension 3)
BLM Sudoku—Advanced (p. D-124, see Extension 3)
BLM Always, Sometimes, Never (p. D-125, see Extension 4)
NOTE: Recognizing and using counterexamples is a mathematical practice (MP.3) that needs
to be explicitly taught. Students will use what they learn in this lesson to solve problems in
other lessons.
Introduce the term counterexample. Draw on the board:
All circles are shaded.
Have a volunteer identify which circle shows that the statement is not true. (the white circle)
Draw on the board:
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-49
Repeat with two different statements about the picture: All squares have a horizontal side. (third
from the left square) All squares are striped. (far left square) SAY: An example that proves a
statement false is called a counterexample to the statement. In other words, a counterexample
shows that the statement is not true.
Exercises:
(MP.3) 1. Which shape is the counterexample to the statement?
A.
B.
C.
D.
a) All triangles are shaded.
b) All triangles have a horizontal side.
Bonus: All triangles are isosceles.
Answers: a) C, b) B, Bonus: C
(MP.3) 2. Find a counterexample for the statement.
a) All animals that live in water are fish.
b) All animals with two legs are humans.
Sample answers: a) whales are mammals that live in water, b) birds have two legs and
are not human
Bonus: Make up your own false statement, and have a partner find a counterexample. Make
sure the statement is true in some cases so that your partner needs to work to find a
counterexample. For example, “All people in my class wear glasses” is a better statement than
“All people in my class are aliens.”
Recognizing when a statement does not apply to all examples. Write and draw on
the board:
All circles are shaded.
OR
If the shape in the picture is a circle, it is shaded.
Explain that the sentences mean the same thing, but they use two different ways to say it.
ASK: What are these statements about? (circles) Underline all the circles. Emphasize that the
statements refer only to the circles; whether any of the other shapes are shaded or not is
irrelevant. ASK: Are all circles shaded? (no) Have a student circle the counterexample. (E)
Erase the underlining and the circling and repeat with the statements below, starting by first
underlining the relevant shapes. Emphasize in each case that the sentence is only about the
shapes you underline; the shapes that are not underlined are not relevant.
All squares in the picture are big. (underline squares, counterexample: D)
All squares in the picture are shaded. (underline squares, counterexamples: D, F)
All big squares in the picture are shaded. (underline large squares, counterexample: F)
All small circles in the picture are shaded. (underline small circles, counterexample: E)
D-50
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Exercises:
1. Use the picture of circles and squares above. List the shapes the statement is talking about.
Then name the counterexample for the statement.
a) All shaded shapes are circles.
b) All white shapes are small.
c) All small shapes are white.
d) All small white shapes are squares.
Answers: a) shaded shapes: A, B, C, counterexample: B; b) white shapes: D, E, F,
counterexample: F; c) small shapes: C, D, E, counterexample: C; d) small white shapes: D, E,
counterexample: E
2. Use the picture of circles and squares above. Name the counterexample for the statement.
a) All shaded shapes are big.
b) All white shapes are squares.
c) All big shapes are squares.
d) All big shapes are shaded.
Answers: a) C, b) E, c) A, d) F
3. Name the counterexamples from the list.
a) All numbers are either positive or negative.
9, −4, 9.6, 1/2, 0, −3.82
b) All right triangles are isosceles.
A.
B.
C.
D.
c) If you add two numbers, the sum is larger than both addends.
3+4
5+0
6 + (−2)
0.4 + 1.2
Answers: a) 0; b) B, D; c) 5 + 0, 6 + (−2), −4 + 9
E.
−4 + 9
Review the word “vowel” if necessary. The letters a, e, i, o, u, and sometimes y are vowels.
Have students signal the answer to the following exercises with thumbs up for yes and thumbs
down for no.
Exercises: Is “Tom” a counterexample for the statement?
a) All names have two vowels.
b) All names have three letters.
c) All names have four letters.
d) All boys’ names start with D.
e) All names are boys’ names.
f) All names read the same backward and forward.
Answers: a) yes, b) no, c) yes, d) yes, e) no, f) yes
Finding counterexamples.
Exercises:
1. Find a counterexample to both statements in the previous exercises for which “Tom” is not a
counterexample. Find one example that works as a counterexample to both statements at the
same time.
Sample answer: Sara
Bonus: Explain why there cannot be a counterexample to all six statements. Hint: Look at parts
d) and e).
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-51
Answer: A counterexample to part d) has to be a male name. It cannot be a counterexample to
part e) because the counterexample to that cannot be a boys’ name. Therefore, there cannot be
a counterexample that disproves all six statements at the same time.
Explain that, in geometry, you often need to show a counterexample to explain why the
statement is false. A statement is often true about some shapes and not true about other
shapes. For example, the statement “In a parallelogram, all sides are equal” is true about some
parallelograms (rhombuses), but not true about all parallelograms. So, for a counterexample, we
need to draw a parallelogram that does not have all equal sides. Write the statement on the
board and invite a volunteer to draw a parallelogram that does not have all equal sides.
Explain that, in the next exercises, students will need to tell what type of shapes they will be
looking at to find a counterexample. For example, you would look at parallelograms to find
a counterexample for the statement about parallelograms. Looking at hexagons would not
make sense.
Point out that two of the parts in the following exercises are exactly the same. They are just
worded differently because geometric statements are often worded in different ways: some of
them look like “all ___ are ____,” and others look like “if___, then____.” You might also point out
that these statements are general statements about types of shapes, but specific shapes can
provide counterexamples.
Exercises: What shapes is the statement talking about?
a) All rectangles are squares.
b) All triangles have at least one right angle.
c) All isosceles triangles have an angle of 60.
d) All triangles with a 60 angle are equilateral.
e) If a triangle has an angle of 60, then it is equilateral.
f) If a quadrilateral has at least two right angles, then it is a rectangle.
g) In all rectangles, the longer side is twice as long as the short side.
h) In all trapezoids, the opposite sides are not equal.
Bonus: If a quadrilateral has two right angles, it is a trapezoid or a parallelogram.
Answers: a) rectangles, b) triangles, c) isosceles triangles, d) triangles with a 60 angle,
e) triangles with a 60 angle, f) quadrilaterals with two right angles, g) rectangles, h) trapezoids,
Bonus: quadrilaterals with two right angles
SAY: To find a counterexample to the statement in part a) in the previous exercises, we need to
look at rectangles. ASK: What must be special about the rectangle for it to be a counterexample
of the statement? (it must be a rectangle that is not a square) SAY: All these sentences have
two parts. When we are looking for a counterexample, we are looking at a shape the sentence
is talking about in the first part, but the second part of the sentence is not true about that shape.
D-52
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Exercises: Describe the counterexample for each statement in the previous exercises.
Answers: a) a rectangle that is not a square, b) a triangle that has no right angles,
c) an isosceles triangle that has no 60 angle, d) a triangle with a 60 angle that is not equilateral,
e) a triangle with a 60 angle that is not equilateral, f) a quadrilateral with two right angles that is
not a rectangle, g) a rectangle where the longer side is not twice as long as the short side,
h) a trapezoid with equal opposite sides, Bonus: a quadrilateral with two right angles but that is
not a rectangle and not a trapezoid
NOTE: If students struggle with the previous exercises, have them circle the two parts in the
sentence first. Give students time to find the counterexamples following the descriptions they
wrote. Encourage students to make a sketch of the shape they are looking for, as shown in the
selected answers below:
f) right trapezoid
h) isosceles trapezoid
Bonus: a quadrilateral that has two right angles that are opposite
True or false? Explain that the next task is going to be harder. You are going to give students
statements that might be either true or false, and they need to decide which ones are true and
which ones are false, and provide a counterexample to the false statements.
Exercises: Is the statement true or false? Provide a counterexample if it is false.
a) All quadrilaterals have sum of interior angles equal to 360.
b) All even numbers have a digit 4.
c) All multiples of 5 have the ones digit 5 or 0.
d) All positive numbers are greater than 1.
e) If a triangle is isosceles, it has at least two equal angles.
f) All quadrilaterals have at least two equal angles.
Bonus: If a parallelogram has a right angle, it is a rectangle.
Answers: a) true, b) false, counterexample: 2, c) true, d) false, counterexample: 0.5 or 1,
e) true, f) false, counterexample:
, Bonus: true
Bonus: For each part, say whether the statement is true or false. Provide a counterexample if it
is false.
If the boxes are filled in with whole numbers, then the answer is a whole number.
a)
+
b)
–
c)
×
d)
÷
Answers: a) true, b) true, c) true, d) false; counterexample: 3, 5
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-53
Extensions
(MP.1) 1. Make up a statement so that the given word or number is a counterexample.
a) the word “run”
b) the number 8
Sample answers: a) all words have 4 letters, b) all numbers are prime
(MP.4) 2. To help students practice making logical deductions, go through the book Anno’s Hat
Tricks by Akihiro Nozaki and Mitsumasa Anno with the class. Work through the book over
several days, a few pages at a time. The book is suitable for Grades 5–12 and students will let
you know when the logic becomes too tough.
(MP.7) 3. Have students complete BLM Sudoku—Warm-Up, BLM Sudoku—Introduction,
and BLM Sudoku—Another Strategy. These BLMs introduce students to sudoku puzzles,
which require substantial logical thinking to solve. Students can complete BLM Sudoku—
Advanced for an added challenge.
4. Explain that a whole number is divisible by another whole number if the answer in the division
is a whole number. For example, 12 is divisible by 1, 2, 3, 4, 6, and 12. Have students complete
BLM Always, Sometimes, Never, which provides practice with logical thinking.
D-54
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-9
Congruence
Pages 93–95
Standards: 8.G.A.2
Goals:
Students will identify congruent shapes with emphasis on triangles.
Students will use equality between corresponding angles and sides to find angles and sides of
congruent triangles.
Students will write congruence statements.
Prior Knowledge Required:
Can measure angles and sides of polygons
Is familiar with notation for equal sides and angles
Can name angles and polygons
Is familiar with the symbol for angle
Can classify triangles
Vocabulary: congruence statement, congruent (), counterexample, isosceles, scalene
Materials:
paper shapes for demonstration
tracing paper for demonstration
scissors
overhead projector
rulers
protractors
transparency of BLM 1 cm Grid Paper (p. I-1)
Introduce congruent shapes. SAY: Congruent shapes have the same size and shape, so if
you put one shape on top of the other, they should match exactly. Explain that you might need
to flip or turn the shapes to make them match.
Attach the shapes below to the board:
SAY: These shapes are congruent. I can turn the shape on the left a quarter turn clockwise and
place it on top of the other shape to check. Demonstrate this with the shapes on the board.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-55
SAY: Now we can see that they match exactly. Move the shapes apart and place them on the
board as shown below:
ASK: What do I need to do with the second shape to get it to match the first shape? (turn it
180 clockwise or counterclockwise and move over the first shape) Invite a volunteer to check.
Repeat with the pair of shapes shown below, which require a flip through a vertical line instead
of a turn:
Attach the shapes below to the board:
ASK: Is this new shape, at right, congruent to the others we have just seen? (no) Why not?
(both of the shapes are made from a row of 4 squares with another square attached to the side,
but in the shape on the left, the additional square is attached to one of the two middle squares;
in the shape on the right, it is attached to the end square.)
Explain that the only things that matter in congruence are size and shape. Position, color,
pattern, and thickness of lines do not matter.
For the exercises below, draw the four shapes for each part on a grid. Students can signal
their answers by raising the number of fingers that corresponds to the correct answer.
NOTE: Students might use shading as a clue. Remind them that shading does not matter.
Exercises: Which shape is congruent to the top shape, 1, 2, or 3?
a)
b)
1.
D-56
2.
3.
1.
2.
3.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
c)
1.
2.
3.
Answers: a) 1, b) 2, c) 2
For each of the previous exercises, ask students to describe informally how they can get the
congruent shape from the original shape. (sample answers: a) rotate 90 counterclockwise,
b) rotate 180 clockwise or counterclockwise or flip upside-down, c) rotate 180 clockwise
or counterclockwise)
Congruent triangles have sides of the same length. Explain that, for the rest of the lesson,
we will be dealing with triangles. Draw on the board:
Triangle 1
Triangle 2
B
7 cm
A
4 cm
C
5 cm
Triangle 3
H
E
4 cm
D
5 cm
6.4 cm
4 cm
7 cm
F
I
J
5 cm
ASK: Which two triangles are congruent? Have students signal the answer. (1 and 2) Use
tracing paper to copy Triangle 2 and cut the tracing out. Show how the cutout matches Triangle 1
exactly. Point out that you need to flip the triangle horizontally to get from 2 to 1. SAY: These
two triangles have sides of the same length: both have a side that is 4 cm long, a side that is
5 cm long, and a side that is 7 cm long. ASK: How is the third triangle different from the other
two triangles? (Triangle 3 does not have a side 7 cm long; Triangles 1 and 2 are obtuse
triangles, but Triangle 3 seems to be a right triangle) Emphasize that to explain why two
triangles are not congruent, you can say that one of them has a side of a different length than
the other. Congruent triangles have to have sides that are the same lengths.
Remind students that they can use the same number of hash marks to show line segments
(such as sides) of equal length. Invite a volunteer to mark the sides that are the same length in
Triangles 1 and 2 with the same number of hash marks. Ask students to write which side in
Triangle 1 is equal to which side in Triangle 2. (AB = EF, BC = ED, AC = FD) Keep the triangles
on the board for later.
NOTE: Even though it is good practice to use the same order for corresponding vertices in the
triangles (so AB = FE, not AB = EF in the example), the equality described above is between
line segments, so writing the vertices in the opposite order is not a mistake. Model the good
practice for students, but there is no need to emphasize it at this point of the lesson.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-57
Exercises: The triangles are congruent.
a) Sketch the triangles. Mark the equal sides with hash marks.
E
B
i)
ii)
iii) W
D
F
H
A
C
K
M
X
Y
V
U
L
I
G
Z
b) Write which side in one triangle is equal to which side in the other triangle.
Selected answers: b) i) DE = GH, DF = GI, EF = HI; ii) AB = KL, BC = LM, AC = KM;
iii) UV = ZY, VW = YX, UW = ZX
To check that the equal sides are identified correctly, explain that you can imagine placing one
of the triangles on top of the other so that the sides match. ASK: What do you need to do with
triangle ABC to get triangle KLM? (flip it upside-down) What side will be placed on top of what
side? (AB on top of KL, BC on top of LM, AC on top of KM) Repeat with the other two pairs of
triangles, which need, in part i), a slide to place GHI on top of triangle DEF, and in part iii),
a 90 rotation clockwise to get from UVW to XYZ.
Congruent triangles have angles of the same size. Return to the earlier example of
Triangles 1, 2, and 3. Demonstrate using the cutouts that Triangles 1 and 2 have angles that
match exactly. Remind students that equal angles are marked with the same number of arcs.
Invite a volunteer to mark the equal angles in triangles ABC and DEF. (A = F, B = E,
C = D)
Exercises: The triangles are congruent.
a) Sketch the triangles. Mark the equal angles with matching arcs.
E
i)
ii) W
iii)
V
D
C
Y
F
B
A
M
H
I
K
J
U
Z
X
L
b) Write which angle in one triangle is equal to which angle in the other triangle.
Answers: b) i) D = I, E = H, F = J; ii) U = X, V = Y, W = Z; iii) A = K,
B = M, C = L
For the pairs of triangles in the previous exercises, discuss strategies for finding the equal
angles. Concentrating on the size of the angles is one strategy. Students should also try to
imagine putting the triangles one on top of each other. Discuss what needs to be done to one
triangle to put it on top of the other: in part i), slide triangle IHJ on top of triangle DEF, in part ii),
rotate triangle XYZ 90 clockwise and shift it to place it on top of triangle UVW, in part iii),
flip triangle KLM horizontally and shift it up to place it on top of triangle ABC.
D-58
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Introduce the triangle and congruence symbols. SAY: We know that triangles UVW and
XYZ in the exercises you have just finished are congruent. Write on the board:
Triangle UVW is congruent to triangle XYZ.
Remind students that mathematicians often use symbols to shorten the notation. Explain that
there is a special symbol for triangles, just as there is a special symbol for angles. SAY: Instead
of writing “triangle ABC,” we can write ABC. Rewrite the sentence on the board:
UVW is congruent to XYZ
Explain that we also have a special symbol we can use to show congruence. Draw “” on
the board. SAY: Just as we can write AB = CD instead of “AB is equal to CD,” we can write
ABC  EFG instead of “ABC is congruent to EFG.” Point out to students that this symbol is
similar to the “approximately equal to” sign (≈).
(MP.6) Writing congruence statements. Explain that the congruence sign means more than
the equal sign does. SAY: Mathematicians have agreed to write congruence statements with
the congruence symbol and in an order that shows the matching sides and matching angles in
the congruent shapes.
ASK: If you want to place triangle UVW on top of triangle XYZ, what would you need to do?
(rotate triangle UVW 90 counterclockwise) SAY: This will help us to write the congruence
statement. Write on the board:
UVW  ___ ___ ___
ASK: Which vertex will U be placed on top of? (vertex X) Point out that when students wrote the
angle equalities, they wrote U = X, so this matches their earlier work. Write “X” in the first
blank. Repeat with the other two vertices to get the congruence statement UVW  XYZ.
Repeat with the pair of triangles from part iii) in the previous exercises. (ABC  KML)
Emphasize that when writing the congruence statement, we must write the letters for the second
triangle exactly in the order of vertices that shows congruence to the first triangle.
(MP.6) Exercises: Write a congruence statement for the triangles.
a)
b) Q
E
D
F
A
M
S
T
P
U
R
N
Answers: a) DEF  ANM, b) PQR  TUS
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-59
Identifying equal elements from congruence statements. Write on the board:
ABC  WUT
SAY: This congruence statement gives me all the information I need to say which sides and
angles are equal in this pair of triangles. Have a volunteer sketch two identical scalene triangles
on the board and label one of them ABC. ASK: According to the congruence statement, which
angle in WUT is equal to A? (W) Label the angle in the second triangle. Continue with the
rest of the angles and invite volunteers to label the angles in the second triangle. ASK: From the
congruence statement, which side in WUT is equal to side AB? (WU) How do you know?
(A, B and W, U are the first two letters in the triangles in the congruence statement) Record the
equality on the board. ASK: Does this fit the sketch? (yes) Continue with the other two pairs of
sides. (BC = UT, AC = WT)
(MP.7) Exercises: Use the congruence statement RAT  COB to write the equalities
between the sides and the angles of the triangles.
Answers: RA = CO, AT = OB, RT = CB, R = C, A = O, T = B
Identifying equal sides and writing congruence statements. Have students copy the three
triangles in the exercises below in their notebooks. Point out that the triangles look very similar
and might be congruent. Remind students that to check if the triangles are congruent, students
need to imagine placing the triangles one on top of the other, trying to make them match
exactly. Then they should measure the sides and the angles of the triangles and see if the
matching sides are equal and if the matching angles are equal. If some of the elements do not
match, the triangles are not congruent. This is similar to looking for a counterexample: if you find
just one pair of sides or angles that should be equal but are not equal, this is enough to say that
the shapes are not congruent.
Exercises: Which triangles are congruent? Use a ruler and a protractor to check. Write the
congruence statement.
O
T
A
R
C
M
B
N
E
Answers: ARC  BNE
Bonus: Draw two congruent isosceles triangles and label them so that OWL  TAC. Write
another congruence statement for the same triangles.
Sample answer: OWL  CAT
L
O
W
D-60
T
C
A
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
(MP.7) Using congruence statements to find sides and angles of congruent triangles.
Draw on the board:
UVW  DEF
W
V
20 cm
30
34.6 cm
40 cm
U
Have students sketch a similar diagram in their notebooks. SAY: These triangles are congruent.
This means that they have angles of the same size and sides of the same length. I would like to
find the size of the angles and the lengths of the sides in both triangles. ASK: What do we know
from the congruence statement? (UV = DE, VW = EF, UW = DF, U = D, V = E, W = F)
Have a volunteer write the equations on the board. Ask students to label the second triangle
using the equations. SAY: I need to turn the second triangle 90 clockwise to get it to the same
position as the first triangle. Redraw the second triangle as shown below, and have students
check that their answers are correct.
W
E
V
F
E
20 cm
40 cm
F
30
34.6 cm
D
U
D
Ask students to transform information they know from both diagrams to the new picture of
triangle DEF. Students should be able to label all the side lengths and two of the angles.
(UV = DE = 40 cm, VW = EF = 20 cm, UW = DF = 34.6 cm, U = D = 30, W = F = 90)
ASK: How can we find the size of the third angle in each triangle? (the angles add to 180, so
the third angle in both triangles, V and E, is 180 − (90 + 30) = 60) PROMPT: What is the
sum of the angles in a triangle? (180)
Extensions
(MP.6, MP.7) 1. Write a congruence statement for the quadrilaterals.
B
M
C
A
D
K
L
N
Answer: ABCD  NLKM
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-61
(MP.1) 2. Draw two non-congruent figures with:
a) the same perimeter
b) the same area
Bonus: the same perimeter and the same area
Sample answer: Bonus:
c) the same shape
(MP.1, MP.3) 3. In a triangle ABC, the sides AB and AC are equal.
a) Sketch the triangle.
b) Mark point D on the side BC so that BD = CD. Draw the line segment AD.
c) The perimeter of ABC is 48 cm. The perimeter of ABD is 36 cm. Find the length of the line
segment AD.
Solution:
A
B
D
C
The perimeter of ABC is AB + BC + AC = 48 cm.
Since AB = AC, and BC = 2 × BD,
AB + BC + AC
= 2 × AB + 2 × BD
= 48 cm
So AB + BD = 24 cm.
The perimeter of ABD = AB + BD + AD = 36 cm
24 cm,
So AD = 36 cm − 24 cm = 12 cm.
D-62
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-10
Supplementary and Vertical Angles
Pages 96–98
Standards: 8.G.A.1b, 8.G.A.5
Goals:
Students will identify supplementary and vertical angles, investigate their properties, and use
these properties to solve problems.
Prior Knowledge Required:
Can measure angles and sides of polygons
Is familiar with notation for equal sides and equal angles
Can name angles and polygons
Is familiar with the symbol for angle
Can classify triangles
Vocabulary: adjacent angles, intersect, intersection point, straight angle,
supplementary angles, vertical angles
Materials:
transparency
overhead projector
Introduce supplementary angles. SAY: Supplementary angles are a pair of angles that add to
180°. If A and B are supplementary angles, we can say that A supplements or is a
supplementary angle to B. Have students signal the answers to the following exercises by
showing thumbs up for yes and thumbs down for no.
Exercises: Are the angles supplementary?
a)
b)
45
c)
63
135
129
115
51
d)
e)
35
155
148
42
f)
83.5
96.5
Answers: a) yes, b) no, c) yes, d) no, e) no, f) yes
Explain that sometimes we can label angles with a single small letter inside the angle. SAY: This
does not work when there are many angles or when they overlap, but when we are dealing with
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-63
a few angles and they don’t overlap, we can use small letters or numbers. Draw the picture
below on the board and trace each angle separately while naming it:
a c
b d
Remind students that straight angles have arms that make a straight line, and the measure of a
straight angle is 180°. SAY: Angles a and c make a straight angle. Are angles a and c
supplementary? (yes) What other pairs of supplementary angles do you see in this picture?
(a and b, d and b, c and d)
Finding the measure of a supplementary angle. Ask students to write an equation that shows
what it means that a and b are supplementary. (a + b = 180°) Mark ∠a in the picture as
115° and have students rewrite the equation using the measure of a. (115° + b = 180°)
ASK: What is the measure of b? (180° − 115° = 65°) Mark the measure in the picture and
have students find the measure of c the same way. Then challenge them to find the measure
of ∠d and to explain the solution. (d = 115) Keep the pictures from the following exercises on
the board for use later in the lesson.
Exercises: Find the measures of the missing angles.
a)
b)
f
e
c
d = 123
b
a
c = 24
d
c)
g
k
h
m = 86
Answers: a) a = c = 57, b = 123; b) f = d = 156, e = 24; c) h = k = 94,
g = 86
Introduce adjacent and vertically opposite angles. Draw on the board:
a
b
Explain that two angles that share an arm, or have an arm in common, such as angles a and b,
are called adjacent angles. Ask students to identify pairs of adjacent angles in one of the
pictures from the previous exercises.
Explain that, when two lines intersect, as in the pictures in the previous exercises, the angles
that are not adjacent and that are formed by the intersecting lines are called vertically opposite
angles, or vertical angles for short. ASK: In the picture for part a), what angle and angle d form a
pair of vertical angles? (b) SAY: We can also say that angle b is vertical to angle d and angle
d is vertical to angle b.
D-64
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Exercises:
1. Are the angles adjacent or vertical?
1 2
6 3
5 4
a) 1 and 2
b) 2 and 5
c) 3 and 6
d) 1 and 6
e) 1 and 4
f) 4 and 5
Answers: a) adjacent, b) vertical, c) vertical, d) adjacent, e) vertical, f) adjacent
2. Identify the pairs of vertical angles in the pictures in the previous exercises.
Answers: a) a and c, b and d, b) f and d, e and c, c) h and k, g and m
Draw on the board:
a)
b)
c)
1
A
2
3
1 2
Explain that there are no vertical angles in these pictures because there are no intersecting
lines, only rays with a common endpoint. Point at each picture in turn and ASK: Are there
adjacent angles? If so, which angles are adjacent? (a) no, b) yes, all three angles are adjacent
to each other, c) yes, 1 and 2 are adjacent)
(MP.3) Vertical angles are equal. ASK: What do you notice about the measures of vertical
angles? (they are always equal) Have students think of how they can explain why this always
happens. Work through the following proof as a class, using the picture below. Have students
suggest what each next step should be and what to put in each blank.
c
d
b
a
a + b = _____, so a = ______________ (180, 180 − b)
b + c = _____, so c = ______________ (180, 180 − b)
So a = ____ (c)
Using rotation to show that vertical angles are equal. Draw the picture below on a
transparency and project it on the board:
2
1
3
4
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-65
ASK: If I turn the picture, will the angles change? (no) Press a pencil tip to the intersection point
on the transparency to create a pivot for turning the transparency. Rotate the transparency 180°
to show how the rotated image coincides exactly with the original image. Another option is to
use two identical transparencies: keep one fixed and rotate the other. Have students identify the
amount of rotation. (180, half turn) Show the rotation several times if necessary. ASK: Which
angles are rotated onto which angles? (1 becomes 3, 2 becomes 4) What does this tell
us about the sizes of 1 and 3? (they are equal) What about 2 and 4? (they are equal)
Point out that another way to look at the sizes of vertical angles is to think of an angle as the
amount of rotation you need to get from one arm to the other. The amount of rotation you need
to get from the lower arm of 1 to its upper arm is the same as the amount of rotation you need
to get from the upper arm of 3 to its lower arm, because the arms of both angles are parts of
the same lines. To help students see that, draw arrows demonstrating the amount of rotation, as
shown below:
2
1
3
4
Keep the pictures from the following exercises on the board for later use.
Exercises: Find the missing angle measures using vertical angles.
a)
b)
c)
c
b
103
a
21
56
93
d
e
n
51
1
2
87
45
130
Answers: a) a = 103, b = 21, c = 56; b) d = 87, e = 93, n = 51; c) 1 = 130,
2 = 45
Draw on the board:
SAY: I want to know which angle is vertical to the marked angle. I’m only interested in the lines
that show the arms of this angle, so I’m going to highlight them. Do so by making the lines
thicker, as shown below in the diagram on the left. SAY: Now the drawing makes it clear where
the vertical angle is. The angle vertical to the marked angle is a combination of two smaller
angles. Draw an arc on the vertical angle, as shown below in the diagram on the right.
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Exercises: Which two angles, when added together, make the angle that is vertical to the angle
marked with an arc?
a)
b)
c)
4
1
1
2 3
2
3
3
4
2
4
1
5
Answers: a) 2 and 3, b) 2 and 3, c) 3 and 4
Finding angle measures using both supplementary and vertical angles. Return to the
pictures from the exercises on the previous page. SAY: In some pictures we are given more
information than we need to find the angle measures. Let’s look at part a). Erase the 56 and
replace it with the label d. ASK: How can we find the measure of this angle from the other two
angles we are given? (180 − 103 − 21 = 56) How do you know? (angle d and the two given
angles make a straight angle, so they all add to 180) Repeat with 93 in part b).
SAY: In part c) we have several angles we do not know. Add the labels 3 to 7 to the unmarked
angles, as shown below:
7
6
4
45
2
5
3
1
130
SAY: We found the measures of angles 1 and 2. ASK: If you know the measure of angle 1, how
can you find the measure of angle 3? (subtract the measure of angle 1 from 180) How do you
know? (angles 1 and 3 are supplementary angles) What is the measure of 3? (180 − 130
= 50) Repeat with angles 4 and 5. (4 = 5 = 135) Then ask students to find the measure of
angle 6 and explain how they know. (180 − (45 + 50) = 85) PROMPT: Are some of the
angles here angles in a triangle? (angles 2, 3, and 6) What do they add to? (180)
Exercises: Find the measures of all angles in the picture using the given angle measures.
a)
b)
c)
45°
58°
154°
95°
112°
Answers:
a)
135 45°
45 135
112
68 68
112°
121°
b)
c)
58°
32
58° 90 32
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
26
154
121 59
154° 26 59 121°
95
85
85
95°
D-67
Extensions
1. Find x.
x
2x
Answer: x + 2x = 180, x = 60
(MP.8) 2. Imagine n lines that all pass through the same point. Label the angles in order,
clockwise, from 1 to 2n. What angle is vertical to 1? Hint: Try different values of n and
make a sketch.
Solution:
3
4
5
Number of lines
n
Sketch
6
5
Angle vertical
to 1
1
4
2
3
7
6
4
8 1
5 4
5
(MP.2) 3. Write an equation and solve for x.
a)
b)
2x
3x
4x + 15°
9
8
2
3
10 1 2
3
4
7 6 5
6
c)
x
3x
60°
n + 1
x
5x
Answers: a) 2x + 3x = 4x + 15, 5x = 4x + 15, so x = 15°; b) 3x = x + 60, 2x = 60, so x = 30°;
c) x + 90 = 5x, 90 = 4x, so x = 22.5°
D-68
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-11
Congruence Rules
Pages 99–101
Standards: 8.G.A.2
Goals: Students will develop and use rules for congruence of triangles.
Prior Knowledge Required:
Can measure angles and sides of polygons
Is familiar with notation for equal sides and equal angles
Can name angles and polygons
Is familiar with the symbols for angle, triangle, and congruence
Can identify congruent triangles
Can write a congruence statement for two triangles
Knows that the sum of the angles in a triangle is 180
Can classify triangles
Vocabulary: angle-side-angle (ASA), congruence rule, congruence statement, congruent,
conjecture, corresponding angles, corresponding sides, corresponding vertices,
counterexample, isosceles, side-angle-side (SAS), side-side-side (SSS)
Materials:
BLM Investigating Congruence (pp. D-126–127, optional)
straws of different lengths and 2 pipe cleaners for each student (optional)
scissors (optional)
The Geometer’s Sketchpad® (optional)
BLM Congruence Rules on The Geometer’s Sketchpad® (pp. D-128–130, optional)
Corresponding sides and angles. Remind students that, when we want to check whether
shapes are congruent, we ask ourselves if we could place the shapes one on top of the other so
that they match exactly. SAY: When we place the shapes one on top of the other, the sides of
the different shapes that will sit on top of one another are called corresponding sides. The
vertices that will sit one on top of the other are called corresponding vertices. And the angles of
the different shapes that will sit one on top of the other are called corresponding angles. If the
shapes are congruent, corresponding sides and angles will be equal. Draw on the board:
SAY: These two triangles are congruent. Some of the equal sides are marked, and a pair of
equal angles is also marked. ASK: If we imagine placing one triangle on top of the other, what
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-69
do we need to do so they will match? (turn the top triangle clockwise a little) Are the sides
marked with thick lines corresponding sides? (no) Point to different sides of the triangle on the
bottom and ASK: Does this side correspond to the thick side of the triangle on the top? Have
students signal yes or no. Students can also signal the answers to the exercises below. After
each question, have a volunteer explain the answer.
Exercises: Are the two thick sides corresponding sides?
a)
b)
c)
Answers: a) yes, b) no, c) yes)
Introduce the idea of congruence rules. SAY: Congruent triangles have three equal
corresponding sides and three equal corresponding angles. However, we do not always need to
check all six pairs of elements to decide that two triangles are congruent. Today we will be
looking for shortcuts—ways to check fewer pairs of angles and sides. Draw on the board:
22
45
22
45
ASK: If we check two pairs of angles and find that these two pairs of angles are equal, do we
need to check that the third pair of angles is equal? (no) How do you know? (the three angles of
a triangle always add to 180) What is the measure of the third angle in both triangles? (113)
SAY: This means we do not have to check all six elements of a pair of triangles; checking three
sides and two angles will be enough. Now let’s see if we can check even fewer.
SSS, SAS, ASA rules. Have students investigate one of the congruence rules (side-side-side,
side-angle-side, or angle-side-angle) in Activity 1 using The Geometer’s Sketchpad®.
Alternatively, have students investigate all three rules using BLM Investigating Congruence.
Activity 1
Use The Geometer’s Sketchpad® for this activity.
Divide students into groups of three. Students will work on the construction individually, each
using one page of BLM Congruence Rules on The Geometer’s Sketchpad®, sharing the
results with the group. Students should tell which elements of the construction could be
modified. (For example: I could modify the first triangle any way I want by moving any of the
vertices. I could only move around the second triangle by moving vertex E, and I could only turn
the triangle by moving vertex D. When I tried to move vertex D, it would only go along a circle,
because it was constructed so that ED has a fixed length.)
Students might need to reflect the triangles they created during the activity to place them on top
of each other. In this case, have students place the triangles so that they share a side and look
like mirror images of each other. Then have students select the common side and use the
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Transform menu option to declare the side a mirror line; they need to choose the option “Mark
mirror” in the Transform menu options. Now if they select one of the triangles and reflect it using
the Transform menu options, they will be able to see that the triangles match exactly.
In their groups of three, have students match each BLM with the congruence rule it seems to be
showing—side-side-side (SSS), side-angle-side (SAS), or angle-side-angle (ASA). Have them
label the BLM with the full name of the congruence rule. For example, page 2 shows the sideangle-side (SAS) rule by keeping two side lengths and the angle between them constant, which
forces the triangle to be fixed.
(end of activity)
Summarize the congruence rules on the board (as in the “Congruence Rules for Triangles” box
on AP Book 8.1 p. 99). Emphasize that the order of elements in the congruence rules is
important: in the side-angle-side (SAS) rule, the equal angles have to be between the
corresponding equal sides; in the angle-side-angle (ASA) rule, the equal sides have to be
between the corresponding equal angles.
Exercises: Identify the congruence rule that tells you that the triangles are congruent.
B
S
E
H
a)
b)
c)
F
I
X
Y
K
A
C
V
d)
e)
O
J
L
G
D
M
R
U
S
A
f)
Z
A
G
P
T
U
L
W
N
Q
Answers: a) ASA, b) SAS, c) ASA, d) SAS, e) SSS, f) ASA
B
T
O
D
Remind students that, in a congruence statement, the corresponding vertices match. So for
example, in part a) above, the congruence statement is ABC  FDE, because if you try to
turn triangle DEF and place it on top of triangle ABC, vertex A corresponds to vertex F
(and A = F), vertex B corresponds to vertex D (and B = D ), and vertex C corresponds to
vertex E (and C = E).
Exercises
(MP.6) Write the congruence statements for the pairs of triangles in parts b)–f) of the
previous exercises.
Answers: b) GHL  JKI, c) BAT  DGO, d) UVW  POQ or UVW  PQO,
e) LMN  TSR, f) USA  YZX
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-71
Bonus: In part d) above, write two different congruence statements with the letters in the first
triangle in the same order. Can you do this for another pair of triangles in this exercise? Draw
another pair of triangles for which you can do this and write two congruence statements.
Sample answer:
UVW  POQ and UVW  PQO
No, you cannot write a statement like for another pair of triangles in the exercise.
ABC  DEF and ABC  FED
B
A
E
C
D
F
Using congruence rules to show congruence. Draw and write on the board:
B
E
A
A = D = 45
C = F = 70
BC = EF = 30 cm
C
D
F
Ask students to sketch the triangles in their notebooks and label the equal sides and angles in
the triangles. Have a volunteer label the equal angles and sides on the board, as shown below:
(MP.6) ASK: Do you think these triangles could be congruent? (yes) Is there a congruence rule
that tells us that these triangles are congruent based on what we know now? (no) Why not?
(because the equal sides in the triangles are not between the corresponding equal angles)
PROMPT: What elements are marked as equal in each of these triangles? (two angles and
a side) Is the side between the corresponding equal angles? (no) SAY: So this situation does
not fit the angle-side-angle (ASA) congruence rule.
ASK: Is the order of equal, or matched, elements the same in both triangles? (yes, the equal
sides are BC = EF = 30 cm, the angles opposite those sides match: A = D = 45, and there
is another match: C = F = 70) If you try to place triangle DEF on top of triangle ABC to
make the triangles match, would the equal sides fall one on top of the other? (yes) What about
the angles? (yes)
SAY: So these triangles have two pairs of corresponding equal angles and one pair of
corresponding equal sides. I would like to use a congruence rule to show the triangles are
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
congruent, but we can’t use the angle-side-angle (ASA) rule yet because the equal sides are
not between the corresponding equal angles. Maybe we can deduce some more information to
see if the rule applies. ASK: What pair of sides would we need to know are equal to use the
ASA congruence rule? (AC, DF) Do we know that these sides are equal? (no) What pair of
angles would we need to know are equal to use the ASA congruence rule? (B = E)
ASK: How can you find the measure of B from the rest of the angles of the triangle? Have
students write down the expression for the measure of the angle. (B = 180° − (45 + 70) = 65)
Repeat for E. (E = 180° − (45 + 70) = 65) ASK: Are B and E equal? (yes) Write on
the board:
B = E = 65
Mark angles B and E as equal on the picture and ASK: Can we now use a congruence rule?
(yes) Which rule? (angle-side-angle, ASA) Invite a volunteer to circle the equalities between
sides and angles that allow us to use the ASA rule. ASK: So, based on this additional
information, can we say that the triangles are congruent? (yes)
(MP.3, MP.6) Exercise: Explain why the triangles ABC and KML are congruent.
A = K = 35
C = L = 120
AB = KM
Answer: B = 180 − (35 + 120) = 25, M = 180 − (35 + 120) = 25, so B = M.
B = M, A = K, AB = KM, so with the ASA rule, the triangles are congruent.
Two pairs of equal angles and one pair of equal sides do not always mean that the
triangles are congruent. Draw on the board:
Q
E
P
R
D
F
P = D = 90
Q = E = 56
PQ = DF = 36 cm
(MP.3, MP.6) ASK: Do these triangles have two pairs of equal angles? (yes) Do they have a
pair of equal sides? (yes) Can we apply one of the congruence rules? (no) Why not? (answers
will vary, but students will likely point out that the triangles do not look congruent) Are the third
angles equal in these triangles? (yes) How do you know? (in both triangles the size of the third
angle is 180 − (90 + 56) = 34) Invite a volunteer to label the equal angles and sides in the
triangles. ASK: Why can we still not use congruence rules here? (in triangle PQR, the side PQ
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-73
is opposite R = 34, but in triangle DEF the side DF is opposite E = 56) PROMPT: The equal
side is opposite one of the angles in both triangles. Which angle? (in triangle PQR, the side PQ
is opposite R, in triangle DEF the side DF is opposite E) What is the size of the opposite
angles in each triangle? (R = 34, E = 56) Emphasize that this means the order of equal
sides and equal angles is different in these two triangles, so the congruence rule does not
apply. SAY: And indeed, the triangles do not look congruent at all! They have the same shape,
but they are obviously different sizes.
(MP.3, MP.6) Exercise: Explain why the ASA congruence rule cannot be used for these
two triangles.
B
Y
C
A
A = X = 40
B = Y = 85
AB = XZ = 3 cm
Z
X
Answer: In triangle ABC, the given side is between angles A and B, which have the measures
40 and 85. In triangle XYZ, the side XZ that is equal to AB is between X = 40 and
Z = 180 − (40 + 85) = 55. The order of equal pairs of angles and equal sides is different in
the two triangles, so the ASA rule cannot be applied. The triangles are not congruent.
Bonus: Use a ruler and a protractor to draw ABC and XYZ with the measurements given.
AAA is not a congruence rule. Write on the board:
If two triangles have three pairs of corresponding equal angles,
then the triangles are congruent.
True or false?
(MP.3) Have students vote on the question above. (the statement is false) PROMPT: Think of
the triangles you saw in the previous exercise. Have students draw a counterexample to the
statement. Students who wish and have time could draw more than one counterexample.
SSA is not a congruence rule. SAY: We checked what happens with three pairs of angles,
three pairs of sides, two pairs of angles and one pair of sides, but we did not check what
happens when two triangles have two pairs of corresponding equal sides and one pair of
corresponding equal angles. Let’s see if the order matters in the side-angle-side (SAS)
congruence rule. Draw on the board:
E
B
A
D-74
C
D
G
F
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
(MP.3) SAY: I drew a copy of triangle ABC and called it DEF. Then I drew an isosceles triangle
EFG, so that EG = EF. Now look at the triangles ABC and DEG. ASK: Which angles are equal?
(A = D) Which sides are equal? (AB = DE, BC = EG) Are the equal angles opposite
corresponding equal sides? (yes, A is opposite BC, D is opposite EG) Are the triangles
congruent? (no) Which description identifies the equal elements, in order, in your triangles:
side-angle-side or side-side-angle? (side-side-angle, SSA) Of the two descriptions, which is a
congruence rule and which is not? (side-angle-side, or SAS, is a congruence rule) How do the
triangles I drew help you to decide? (they are a counterexample to this statement) Emphasize
that the order of equal corresponding elements matters in triangles. The pair of equal angles
has to be between the pairs of equal corresponding sides for the triangles to be congruent.
(MP.3) Exercise: On grid paper, draw a counterexample to SSA. Use a ruler.
Sample answer:
Using congruence rules to find sides or angles in congruent triangles. Draw on the board:
W
V
17.3 cm
20 cm
30
T
10 cm
R
30
17.3 cm
S
U
SAY: I want to find the size of the angles and the lengths of the sides in both of these triangles.
Let’s see if these triangles are congruent. If the triangles are congruent, we will know that the
lengths of all the sides in both triangles match even though we haven’t measured them. Have
students sketch the triangles and label all the known information. Then ask them to label the
angles that are the same with arcs and the equal sides with the same number of hash marks.
Point out that we do not know yet if the triangles are congruent, and so we cannot, for example,
mark angles T and V as equal.
ASK: How many pairs of equal sides have you marked? (1 pair, UW = RS) How many pairs of
equal angles do we see? (2 pairs, W = R, U = S) Which congruence rule would we like to
apply? (angle-side-angle, or ASA) Is the pair of equal sides between the corresponding equal
angles? (yes) SAY: Then we can apply the congruence rule ASA. Have students write the
congruence statement and the rest of the angle and side equalities. Then ask them to find the
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-75
lengths of all the sides and angles. (see answers below) Ask volunteers to explain how they
found the answers.
UVW  STR
UV = ST = 20 cm
VW = TR = 10 cm
V = T = 180 − (90 + 30) = 60
Remind students that, when several lines or rays meet at a point and there are overlapping
angles, we use three letters to label an angle with the vertex label always in the middle.
Draw the picture in the exercises below on the board, mark different angles with arcs one at a
time, and ask students to name them.
Exercises: Explain why the triangles are congruent. Then write the congruence statement and
find the missing angle measures.
A = 43
B
D
E = 83
AB = CD = 3 cm
AC = CE = 3.7 cm
BC = DE = 2.5 cm
A
E
C
Answers: The triangles have 3 pairs of corresponding equal sides, so by the side-side-side,
or SSS, rule they are congruent, and ABC  CDE. Therefore, A = DCE = 43,
BCA = E = 83, and B = D = 180 − (43 + 83) = 54.
Extensions
(MP.1, MP.3, MP.7) 1. In the diagram below, AC  BD, AO = CO, and BO = DO.
B
A
O
C
D
a) Copy the diagram and label the equal sides and right angles.
b) Use triangles AOB and COD to explain why AB = CD.
c) Explain why AD = BC.
d) Use triangles AOB and COB to explain why AB = CB.
e) What type of quadrilateral is ABCD?
Selected solutions:
b) Since AC  BD, AOB = COD = 90 and we know that AO = CO and BO = DO. This
means we have two pairs of corresponding equal sides and a pair of corresponding equal
angles between them, so by the SAS rule AOB  COD. Therefore AB = CD.
d) Since AC  BD, AOB = COB = 90 and we know that AO = CO. The side BO is in both
triangles, so we have another equal side in both triangles. This means we have two pairs of
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
corresponding equal sides and a pair of corresponding equal angles between them, so by the
SAS rule AOB  COB. Then AB = CB.
e) ABCD is a rhombus.
(MP.1, MP.3, MP.7) 2. Maria drew 2 right triangles as shown below, with PQ = TR and
QT = RS. She thinks that the points P, T, and S are on the same line. Is she correct? Explain.
Hint: Find the measure of PTS.
P
T
Q
S
R
Solution: Maria is correct. Since PQ = TR and QT = RS and the angles between the
corresponding pairs of equal sides are equal, Q = R = 90, the triangles are congruent by the
SAS rule. This means PTQ = S. From the sum of the angles in a triangle we know that:
S = 180 − (90 + RTS) = 90 − RTS.
PTS = PTQ + QTR + RTS
= S + 90 + RTS
= 90 − RTS + 90 + RTS
= 180
The angle PTS is a straight angle, so the points P, T, and S are on the same line.
(MP.1, MP.3) 3. ABC and DEF are both isosceles triangles. A = D and
AB = DE. Are ABC and DEF always congruent? Explain.
Hint: Make a sketch that includes all the information you have been given. Try making more
than one sketch using a different position for the equal angles.
Answer: The triangles do not have to be congruent. Sample counterexample:
F
B
A
E
D
C
(MP.3) 4. Sketch a counterexample to show why the statement is false.
ABC has AB = BC = 7 cm and DEF has DE = EF = 7 cm. So ABC ≅ ΔDEF.
Sample answer:
E
B
7 cm
7 cm
A
C
D
F
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-77
G8-12
Congruence (Advanced)
Pages 102–103
Standards: 8.G.A.2, 8.G.A.5
Goals:
Students will use informal arguments involving congruent triangles to solve problems.
Prior Knowledge Required:
Can name angles and polygons
Can measure angles and sides of polygons
Is familiar with notation for equal sides and angles
Can classify triangles
Knows that the sum of the angles in a triangle is 180
Is familiar with the symbols for angle, triangle, and congruence
Can identify congruent triangles
Knows the SAS, ASA, and SSS congruence rules
Can write a congruence statement for two triangles
Vocabulary: ASA (angle-side-angle), congruence rule, congruence statement, congruent,
conjecture, corresponding angles, corresponding sides, corresponding vertices,
counterexample, isosceles, midpoint, SAS (side-angle-side), SSS (side-side-side),
supplementary angles, vertical angles
Materials:
BLM Two Pentagons (p. D-131)
scissors
Review congruence rules and properties of isosceles triangles. Remind students that
congruence rules are shortcuts that allow us to determine whether or not triangles are
congruent by checking only three elements (sides or angles). ASK: What three elements could
we use? (3 sides, 2 sides and 1 angle, 1 side and 2 angles) Can we use any three elements?
(no; for example, we can’t use 3 angles) Remind students that the order of the elements is
important. Draw on the board:
Remind students that, in an isosceles triangle, the angles between the equal sides and the third
side are equal. SAY: Both these triangles are isosceles right triangles. ASK: What is the size of
the angles that are not marked? (45) How do you know? (angles in a triangle add to 180, and
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
one angle is 90, so the other two angles add to 180 − 90 = 90; since they are equal, they
measure 90÷ 2 = 45 each) Mark the angles in the triangles on the board as 45.
(MP.3) Point to the triangles and SAY: These triangles have all angles the same size, and they
have some sides that are equal. ASK: Are they congruent? (no) Why not? (they are different
sizes) Why can you not apply any of the congruence rules here? (the sides that are equal in
both triangles are in different places in relation to the equal angles: the equal side is between
the 45° angles in the smaller triangle but between the 90° angle and one of the 45° angles in the
other) Remind students that the equal sides need to be adjacent to the equal angles for the
triangles to be congruent using the side-angle-side (SAS) rule.
ASK: When you have two pairs of equal sides and a pair of equal angles, where does the equal
angle have to be for the triangles to be congruent? (in both triangles, the equal angle has to be
between the 2 pairs of equal sides) Draw on the board:
E
B
A
C
D
G
F
Point out that the two triangles have two pairs of equal sides and a pair of equal angles, but the
triangles are not congruent. SAY: Even though the order of the equal elements is the same in
both triangles—angle-side-side—the triangles are still not congruent because the congruence
rule requires the order side-angle-side.
Remind students that the third congruence rule is the side-side-side rule: if two triangles have all
sides of the same length, they are congruent. Also remind students that a congruence
statement lists the vertices of the triangles so that you can say which angle is equal to which
angle and which side equals which side. For example, if ABC ≅ PQR, then we know that
A = P, B = Q, AB = PQ, and so on.
Exercises: Which congruence rule tells that the two triangles are congruent? Write the
congruence statement.
a) b)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-79
W
Bonus:
V
X
U
Answers: a) ASA, GHI ≅ JLK; b) SAS, MOQ ≅ PRN; c) ASA, ABC ≅ FDE;
Bonus: SSS, UVW ≅ WXU
Identifying congruent triangles and explaining why they are congruent when triangles
have a common side or vertex. Explain that sometimes you see triangles that share a side or
a vertex. Draw on the board:
B
A
D
C
Have students copy the picture. SAY: To prove that triangles ABD and BCD are congruent, you
need to use the fact that the common side BD belongs to both triangles and therefore makes a
pair of equal sides. ASK: What sides are equal in these triangles? (AD = CD, BD = BD) Point
out that we know that AD and DC are equal, so we can describe D as at the midpoint in the line
segment AC—in other words, D divides the line segment in half. Write the equalities between
the sides on the board and have students write them in their notebooks. ASK: What angles are
equal in these triangles? (ADB = CDB) PROMPT: What is the size of angle ADB? (90)
Have students write the equality. ASK: What congruence rule can you apply? (SAS) What do
you need to check to be sure the rule applies? (that the matching angle is between the sides
listed in the equalities) Is this the case? (yes, angles ADB and CDB are between the sides
AD and DB, and CD and DB) Have a volunteer write the congruence statement on the board
and have students write it in their notebooks. (ABD ≅ CBD) Leave this picture on the board
for later use.
Exercises: The pairs of equal sides and angles are marked in the diagram. Which congruence
rule can you use to prove that the triangles are congruent?
a)
b)
c)
d)
Answers: a) ASA, b) SAS, c) SSS, d) SSS
Draw two intersecting lines on the board and remind students which angles are vertical angles,
and that vertical angles are equal.
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
(MP.6) Exercises:
a) Write the equalities between the sides and the angles in the triangles.
B
B
i)
ii)
O
C
A
C
D
iii)
A
D
Bonus:
L
W
Z
O
K
M
X
Y
N
b) Which congruence statement can you use to show the triangles are congruent?
c) Write the congruence statement.
Answers:
a) i) AD = CD, AB = CB, BD = BD; ii) AO = CO, BO = DO, COD = AOB;
iii) KO = MO, KON = MOL, OKN = OML; Bonus: ZY = XW, ZYW = XWY, YW = WY
b) i) SSS; ii) SAS; iii) ASA; Bonus: SAS
c) i) ABD ≅ CBD; ii) ABO ≅ CDO; iii) KNO ≅ MLO; Bonus: YZW ≅ WXY
ASK: In part ii), how do you know the angles COD and AOB are equal? (they are vertical
angles, and vertical angles are equal) What other pair of vertical angles did you see in this
exercise? (KNO and MLO)
Using congruence. Return to the picture below, from earlier in the lesson.
B
A
D
C
AD = CD
BD = BD
ADB = CDB
ABD ≅ CBD
Remind students that they showed that the triangles are congruent using the side-angle-side
congruence rule. Ask students to write the rest of the equalities between the sides and the
angles of the triangles. (AB = BC, A = C, ABD = CBD) ASK: What have we just written
about the larger triangle, ABC? (the triangle is isosceles) To prompt students to see the answer,
trace the larger triangle with a finger and ask students to classify it.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-81
Add the information in the exercises below to the triangles from Exercise 2, above. Students
might need a prompt to identify ADB as a right angle in part a). If so, ask them what the
measure of angle ADC is, and what they know about the two angles with vertex D. When
students have finished their work, discuss solutions as a class.
Exercises: Find all the missing sides and angles that you can.
a)
b)
c)
B
B
L
7 in
5 cm
A
O 60 70
C
64
A
6m
O 51
K
M
C
D
D
N
Answers: a) C = 64, ADB = CDB = 90, BC = 5 cm; b) CD = 7 in, COD = 60,
OCD = 70, OBA = ODC = 50; c) LM = KN = 6 m, KON = MOL = 51,
MLO = KNO = 39
Bonus: Use the picture to prove that, in an isosceles triangle, the angles between the equal
sides and the third side are equal (or A = C).
B
A
C
D
Answer: AB = CB, BD = BD, and ABD = CBD, so by the SAS congruence rule,
ABD ≅ CBD. From the congruence statement we know that C = A, so in ABC,
AB = BC.
Using notation shortcuts in diagrams. Draw the picture below on the board and have
students copy it:
G
F
H
FE = GH
EFH = GHF
E
SAY: We have a quadrilateral EFGH that is made from two triangles with a common side,
another pair of equal sides, and equal angles. I would like to see what properties this
quadrilateral has. ASK: What can you tell about triangles EFH and FGH? (they are congruent)
How do you know? (they have two pairs of equal sides and a pair of equal angles between
them, so by the SAS congruence rule the triangles are congruent) Ask students to write the
congruence statement for the triangles. (EFH ≅ GHF) Then ask them to write the equalities
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
for the sides and the angles that follow from the congruence statement. (EH = GF, FEH = HGF,
EHF = GFH)
Point out that listing all the angle names takes a long time, and it is often hard to see which
angle name is equal to which angle on the diagram. In addition, if you are talking about
several angles, it becomes difficult to use arcs. SAY: In such cases, we often mark the size
of the angles and sides with letters, using the same letter for angles of the same size. Label
EFH = a, FEH = b, and EHF = c, and have students to label all the equal angles on the
diagram with these letters. Have a volunteer do the same on the board. The picture will look like this:
b
F
G
c
a
a
c
H
b
E
SAY: Now it is clear from the picture that the quadrilateral also has equal opposite angles.
Angles E and G are equal, but so are angles EFG and GHE. ASK: Using letters, what are
angles EFG and GHE equal to? (a + c)
ASK: What type of a quadrilateral does EFGH seem to be? (a parallelogram) Point out that
parallelograms have equal opposite sides and equal opposite angles, but students have not
proven that the quadrilateral has parallel opposite sides. The fact that EFGH is a parallelogram
remains a conjecture—something we think is true but have not proved using logic. Explain that
students will be able to prove this conjecture later in this unit.
(MP.3, MP.7) Exercise: Prove that quadrilateral PQRS has equal opposite angles. Use the
shortcut notations for the angles.
Q
P
R
S
Answer: PQ = RS, QR = SP, and PR = PR (common side), so by the SSS congruence rule,
PQR ≅ RSP. From the congruence statement, the corresponding angles in the triangles are
equal, as labeled below.
Q
c
P
a
b
b
a
S
R
c
This means Q = S and QPS = a + b = SRQ, so the opposite angles in the quadrilateral
PQRS are equal.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-83
Bonus: XYZW is a rhombus with line segment XZ.
a) Sketch the situation.
b) Prove that YXZ = WXZ = YZX = WZX.
Answers:
a)
Y
X
Z
W
b) XY = XW, YZ = WZ, XZ = XZ, so by the SSS congruence rule, XYZ ≅ XWZ. From the
congruence statement, YXZ = WXZ and YZX = WZX. Since XYZ is isosceles,
YXZ = YZX, so all angles are equal.
Congruence in other polygons. Remind students that they can talk about congruence of any
pair of polygons. SAY: If you can place two polygons one on top of the other and they match
exactly, they are congruent. Explain that, in polygons with more than three sides, the order of
the equal sides and angles is even more important than in triangles. It is not enough to say that
the shapes have all sides the same lengths and that all angles are the same size; the equal
sides and angles have to match in order.
(MP.3) Activity
Give students BLM Two Pentagons. Have students work individually to cut out the pentagons
and compare the sides and the angles to answer the questions on the BLM. When students are
finished, ask the class Question a) and have them signal the number of right angles on each
polygon to check their answer. (3 each) Read Question b) to the class, and have them hold up
the folded shapes so that they can show that the remaining four angles are all equal. Ask the
class Questions c) to h) and have students to signal their answers to each one so that you can
check the whole class at the same time. Students can signal thumbs up if their answer is “yes”
and thumbs down if their answer is “no.” (c) yes, d) yes, e) no, f) no, g) yes, h) no)
Have students answer Question i). Pair students who are struggling with students who show
greater understanding of the material to compare their answers (the former can coach the latter
as they come up with a common answer). Repeat with groups of four and groups of eight, and
then have the groups share their answers with the whole class. (The order of sides and angles
matters. For example, in Pentagon A the obtuse angles are adjacent, but in Pentagon B the
obtuse angles are separated by a right angle. The order of the equal sides is not the same on
these pentagons, so they are not congruent.)
(end of activity)
Explain that, when we place Pentagons A and B from the BLM one on top of the other, we
define the order in which we will check the angles and sides. If we place the pentagons so that
at least one pair of sides or angles matches, we cannot make all the remaining corresponding
sides and angles match. For example, you can match the shapes so that two angles and the
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
side between them correspond, but the other sides adjacent to the angles are not equal
because the order of the sides in each shape is different.
SAY: When you check for congruence, you either need to try all the possible combinations or
show why congruence is impossible. In this case, congruence is impossible because all the right
angles are adjacent in Pentagon A but not in Pentagon B.
Extensions
1. Look again at the pentagons on BLM Two Pentagons. Which one has greater area? Take the
necessary measurements to check.
Answer: Pentagon A
(MP.1, MP.3) 2. In the quadrilateral ABCD, АВ = CD and ВС = AD. Copy the picture.
Answer the questions to explain why the point O divides the line segments BD and AC in half.
(In other words, explain why O is the midpoint of both AC and BD.)
a) ABD ≅CDB by the _____ congruence rule, so ABD = ________
Label the equal angles with letter a.
b) ABC ≅ CDA by the _____ congruence rule, so BAC = ________
Label the equal angles with letter b.
c) Shade AOB and COD.
d) AOB ≅_________ by the ________ congruence rule, so BO = ______ and
AO = _______, so the point O divides the line segments ______ and ______ in half.
Answers:
a) SSS, CDB; b) SSS, DAC; d) COD, ASA, DO, CO, AC, BD
a
b
b
a
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-85
(MP.4) 3. Tina wants to measure the distance from point X to point Y, but a pond stops her from
walking directly between the points. She finds a point, Z, from which she can walk to both X and
Y in a straight line.
Pond
Y
X
Z
Copy the sketch and follow the steps below to see how Tina solves the problem.
a) Tina walks from X to Z and counts her steps. She continues in a straight line from Z, walking
the same number of steps. She labels the point she stops at W. Draw the point W on the sketch.
Mark the equal distances.
b) Tina repeats the task in part a), walking from Y through Z, to find point U so that YZ = ZU.
Draw the point U on the sketch. Mark the equal distances.
c) Tina measures the distance UW. Explain why this distance is the same as the distance
between X and Y.
Answers:
Pond
X
Y
Z
U
W
c) XZ = ZW and YZ = ZU as constructed. Also, XZY = WZU because they are vertical
angles. So XZY ≅WZU by the side-angle-side (SAS) congruence rule. From the congruence
statement, XY = WU.
D-86
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-13
Exterior Angles of a Triangle
Pages 104–105
Standards: 8.G.A.5
Goals:
Students will discover, prove, and use the fact that the exterior angle in a triangle equals the
sum of the non-adjacent interior angles.
Prior Knowledge Required:
Knows that the sum of the angles in a triangle is 180°
Can identify supplementary and vertical angles
Knows that vertical angles are equal
Can draw and measure with a ruler and a protractor
Can identify and construct parallel lines using a protractor
Vocabulary: conjecture, exterior angle, interior angle, parallel, supplementary angles,
vertical angles
Materials:
protractors
The Geometer’s Sketchpad®
Introduce exterior angles. Draw on the board:
ASK: What do you know about the measures of angles a, b, and c? (they add to 180) What do
you know about angles c and x? (they add to 180) What are angles like c and x called?
(supplementary angles)
Ask students if anyone knows what “exterior” means. (outer, on the outside) Explain that an
angle such as angle x, created by extending one of the sides outside the triangle, is called an
exterior angle of the triangle because it is outside the triangle. The angles inside the triangle are
called interior angles.
Looking for a pattern in the measures of exterior and interior angles. Mark the measures of
a and b in the triangle on the board as 50°and 57°, respectively. Ask students to find the
measure of c. (73°) ASK: How do you know? (180 − (50 + 57) = 73) What do you know
about c and the exterior angle, x? (they are supplementary angles; they add to 180°) Have
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-87
students find the measure of x. (107°) Draw the table below on the board and fill in the first
column with the information from this triangle:
a
b
x
50
57
107
Ask students to each draw a triangle in their notebooks and label the angles a, b, and c. Then
ask them to extend one of the sides of the triangle that make angle c beyond the vertex so that
the exterior angle is x. Have students measure the angles a, b, and x, and write the measures in
the table. Then have them exchange notebooks and repeat the exercise with the triangles
drawn by their peers.
(MP.7) Ask students to look for a pattern in their tables and have them formulate a conjecture
about the sizes of the angles. Have students pair up and to improve the conjecture they have
written, using the words “exterior” and “opposite.” Students can improve their conjecture again
in groups of four. Have all groups share their conjectures with the class. (2 interior angles add to
the measure of the exterior angle that is the supplementary angle of the third interior angle) You
might point out that the angles that add to the exterior angle are opposite the third angle in
the triangle.
The activity below allows students to check their conjecture using The Geometer’s Sketchpad®.
Activity
Use The Geometer’s Sketchpad® for this activity.
Checking that the exterior angle equals the sum of the interior angles opposite to it.
a) Construct a triangle ABC and measure its angles.
b) Draw a ray BC and mark a point D on the ray, outside the triangle.
c) Measure ACD.
d) Using the Number menu option, calculate the sum of the measures of ABC and BAC.
You can click on the angle measures to make them appear in the calculation windows.
e) What do you notice about the answers in parts c) and d)?
f) Modify the triangle. Did your answer to part e) change?
Answers: e) the answers are the same, f) no
(end of activity)
(MP.3) Proving the conjecture for the size of exterior angle. Return to the triangle above
and erase the angle measures. ASK: How can you find the measure of angle c from the
measure of angle x? (c = 180 − x) Write the equation on the board. ASK: How can you find
the measure of angle c from the measure of angles a and b? (c = 180° − (a + b)) Write the
second equation underneath the first, as shown below:
c = 180 − x
c = 180° − (a + b)
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
SAY: The expressions on the left side of these equations are the same. This means the
expressions on the right side are the same too. ASK: How are the expressions on the right side
of the equal sign the same in both equations? (something is subtracted from 180) Have a
volunteer circle the parts that are subtracted. ASK: What can you say about the subtracted
parts? (they are the same) Have students write the equation showing this. (x = a + b)
Point out that students have now proved their conjecture about the external angle using logic.
Summarize on the board:
An exterior angle of a triangle equals the sum of the two angles opposite to it in
the triangle.
x = a + b
Use the measure of the exterior angle to find missing angles. Work through the examples
below as a class. Then have students work individually on the following exercises.
61
38
115
x
x
68
x = 68 + 38 = 106
x = 115 − 61 = 54
Exercises: Find the measure of angle a.
a)
b)
Bonus:
a
70
45
a
32
127
a
47.5
58
Answers: a) 90, b) 82, Bonus: 22.5
SAY: Now you will use what you know about exterior angles, vertical angles, and supplementary
angles to find the missing angle measures. Draw on the board:
d 58
c b
a
15
ASK: Which angles are the exterior angles for this triangle? (c and the angle labeled 58) What
is the measure of angle a? (43) How do you know? (58 − 15 = 43) Have students find the
rest of the angles in the picture, and then have volunteers explain the solutions. (b = 122,
supplementary to 58 or using sum of the angles in a triangle; c = 58, supplementary to b or
vertical to 58; d = 122, supplementary to 58 or vertical to b)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-89
Exercises: Find the measures of angles x and y.
y
x
x
y
52
71
x
72
x
72
41
y
Answers: a) x = 54.5°, y = 109°; b) x = 36°, y = 144°; c) x = 128°, y = 93°
Extensions
(MP.3) 1. Josh thinks that the exterior angle in a triangle is larger than each of the interior
angles not supplementary to it.
a) Make a sketch of a triangle with an exterior angle.
b) Is Josh’s statement true for your triangle?
c) Is Josh’s statement true for any triangle? Explain.
d) Ted thinks the exterior angle is larger than any angle in a triangle. Is Ted’s statement
true? Explain.
Answers: c) Yes, Josh’s statement is true for any triangle. If x is the exterior angle,
x = a + b, all the measures are positive numbers, and the measure of x, being the sum,
is larger than any of the addends—in other words, larger than any two interior angles that can
add to the exterior.
d) Ted’s statement is not true. Counterexample:
a
70
47.5
The angle adjacent to the exterior angle is an obtuse angle, so it is larger than 70; in this case
it is 110 and it is the largest of the three interior angles and larger than the exterior angle.
(MP.3) 2. The angles QPR and PQR in triangle PQR are acute. Point S is on the line PQ so that
RS  PQ. Jenny thinks that point P can be between the points Q and S. Is she correct? Explain.
Hint: Make a sketch placing point P between Q and S. Look at triangle PRS. What can you say
about the size of QPS?
Answer: Jenny is not correct.
R
Q
P
S
If point P is between Q and S, and RS  PQ, triangle PRS is a right triangle with external angle
QPR. The external angle QPR has to be larger than PSR = 90, so QPR is an obtuse angle.
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
However, we are given that angle PQR is acute. An angle cannot be both acute and obtuse, so
point P cannot be between points Q and S.
(MP.3) 3. Remind students that, in an isosceles triangle, the angles between the equal sides
and the third side are equal. Then present them with the following problem.
Grace thinks ABC is a right angle. Is she correct? Explain without measuring.
B
y
A
x
110º
D
C
Solution: Angle C is equal to angle x, so 2x = 110 because ADB is the exterior angle
for triangle BDC. So x = 55. Triangle ABD is an isosceles triangle, so A = y, and
from the sum of the angles in triangle ABD, 2y = 180 − 110 = 70, so y = 35.
ABC = x + y = 55 + 35 = 90. Grace is correct: ABC is a right angle.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-91
G8-14
Corresponding Angles and Parallel Lines
Pages 106–108
Standards: 8.G.A.5
Goals:
Students will use informal arguments to establish facts about corresponding and co-interior
angles at parallel lines.
Prior Knowledge Required:
Can identify parallel lines using a protractor
Can draw and measure with a ruler and a protractor
Can identify supplementary and vertical angles
Knows that vertical angles are equal
Knows what counterexamples are
Vocabulary: co-interior angles, conjecture, corresponding angles, counterexample, parallel,
supplementary angles, vertical angles
Materials:
transparencies
overhead projector
rulers
protractors
The Geometer’s Sketchpad®
Introduce corresponding angles. Draw on the board:
3
5
7
1 2
4
6
8
SAY: We are going to look at different situations when two lines intersect a third line. One
situation is when angles create a pattern like in the letter F. We call these corresponding angles.
NOTE: The term “corresponding angles” here is not the same as the corresponding angles in
congruent triangles—in other words, not the angles that match up in congruent shapes.
SAY: In the case of corresponding angles among intersecting lines, the letter F pattern can be
flipped from side to side or rotated in a circle. For example, angles 4 and 8 in the picture are
corresponding angles. Trace the letter F in the picture with your finger. Then copy the picture
D-92
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
three more times, trace the letter F in each picture, and have students find all four pairs of
corresponding angles, as shown below:
1
3
5
7
1
3
6
5
8
7
1
3
5
7
2
4
6
8
1
2
3
4
5
6
7
8
2
4
2
4
6
8
If students do not see that the upside-down or reflected pattern resembles an F, copy the
picture to a transparency, highlight the letter F, and turn it over or rotate it so that students see
the pattern.
Exercises: List the corresponding angles.
a)
5
1
6
2
3 4
7 8
b)
a
b
c
d
k
m
n u
Answers: a) 1 and 3, 2 and 4, 5 and 7, 6 and 8; b) a and k, b and m,
c and n, d and u
Corresponding angles for parallel lines are equal. Have students draw a pair of parallel lines
by using the opposite sides of a ruler. Ask them to place the ruler across the two lines they drew
and draw a third line intersecting both lines. Have them measure all eight angles created this
way with protractors and identify which angles are corresponding angles. ASK: What do you
notice about the measures of corresponding angles? (they are equal) Did everyone create
angles of the same size? (no) Did everyone get the same result: corresponding angles in a pair
of parallel lines are equal? (yes)
The activity below allows students to check this result with The Geometer’s Sketchpad®.
Activity 1
Use The Geometer’s Sketchpad® for this activity.
Checking that corresponding angles for parallel lines are equal.
a) Draw a line. Label it AB. Mark a point C not on the line and construct a new line parallel to
AB through C.
b) Draw a line through points B and C.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-93
c) On the line parallel to AB, mark point D on the same side of BC as point A.
d) On the line BC, mark a point E outside the line segment BC, as shown below:
not here
here B
C or here
e) Name two corresponding angles in the diagram.
f) Measure the corresponding angles you named. What do you notice? Make a conjecture.
g) Move the points A, B, C, or D around. Are the corresponding angles always equal?
(MP.5) h) Move point E so that the angles you measured stop being equal (e.g., move point E to
be between B and C). Look at the pattern the angles create. Are they corresponding angles?
Does this create a counterexample to your conjecture from c)? Explain.
Answers: f) corresponding angles at parallel lines are equal; g) yes; h) When angles become
not equal, they are not corresponding angles anymore. This does not create a counterexample,
because the statement talks about corresponding angles and the unequal angles are not
corresponding.
(end of activity)
Practice finding measures of corresponding angles at parallel lines. Draw on the board:
x
137
SAY: The lines are parallel. ASK: What do you know about angle x and the angle that measures
137? (they are corresponding angles, so they are equal) What is the measure of angle x? (137)
Exercises: Find the measure of the corresponding angles.
a)
b)
23
70
x
c)
a
Bonus:
x
101
54
y
60
65
y x
Answers: a) x = 70, b) a = 23, c) x = 101, y = 54, Bonus: x = 60, y = 65
D-94
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Find measures of angles between intersecting lines using corresponding, vertical, and
supplementary angles. Review what vertical angles are and the fact that they are equal.
Remind students what supplementary angles are. Then return to part c) in the previous
exercises and work as a class to fill in all the missing angle measures. (see answers below)
79 101
79 x
101 79 101 79
54
126
y
126 54
126
126
54
Exercises: Find all the angles in the picture.
a)
b)
66
73
c)
123
a
Bonus:
c
b
65
60
53
123
y x
d
Answers:
a)
66 11466
b)
73 107 90 90
114 66 114
66 114 66
c)
b
c
123 57 123 57 127 53
a
57 123 57 123 53 127
d
107 73
73 107
90 90
90 90
Bonus:
55
65
60
65
65
55
120
60
115 65 120 60
65 115
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-95
(MP.3) When corresponding angles are equal, lines are parallel. Write on the board:
When lines are parallel, corresponding angles are equal.
SAY: We have seen that this statement is true. Let’s change the order in this sentence.
Continue writing on the board:
When corresponding angles are equal, lines are parallel.
Point out that when we change an order in a sentence, the meaning changes. For example,
the sentence “All boys are people” is true, but the sentence “All people are boys” is not true.
ASK: Do you think the statement on the board is true? You might want to have students vote.
Ask students to draw a pair of lines that are not parallel and then a third line that intersects
them. Have them pick a pair of corresponding angles in their drawing and measure them.
ASK: Did anyone get a pair of equal corresponding angles? (no) Explain that this shows that the
statement “When corresponding angles are equal, lines are parallel” is likely to be true, but does
not prove it. SAY: However, mathematicians have proven that the conjecture is true. Write “true”
beside the statement on the board. NOTE: If your class is ready for the challenge, you can have
them do Extension 1 to prove the statement.
Using corresponding angles to identify parallel lines. Explain that the statement “When
corresponding angles are equal, lines are parallel” allows you to tell which lines are parallel and
which are not. Return to the picture in part c) of the previous exercises and ask students to
circle a pair of corresponding angles for each pair of lines, using different colors. Point out that,
to make identifying lines easier, you used small letters to name the lines in this picture. Then
ask students to identify which lines are parallel. (b and c)
Activity 2
Have students work in pairs to complete this activity.
a) Draw two pairs of lines, one parallel and the other not but looking like it might be. (For example,
draw a line and then place a ruler along the line as if to draw another line along the parallel side
of the ruler. Then rotate the ruler very slightly.) Do not indicate which pair is which.
b) For each pair of lines, draw a third line intersecting the first and second lines.
c) Exchange your paper with a partner. Measure and compare the corresponding angles to
identify the pair of parallel lines.
(end of activity)
Introduce co-interior angles. Explain that, when two lines intersect a third line, there can be
other patterns of angles, which also have special names. Draw on the board:
3
5
7
D-96
1 2
4
6
8
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Explain that angles that create a pattern like in the letter C are called co-interior angles or
same-side interior angles. For example, angles 4 and 6 in the picture are co-interior angles.
Trace the letter C in the picture with your finger. Then copy the picture and have students find
the second pair of co-interior angles. (3 and 5) Trace the letter C in the second picture, as
shown below:
1
3
5
7
2
1
4
3
6
5
8
7
2
4
6
8
In parallel lines, co-interior angles are supplementary. Draw on the board:
138
Point to different angles in the picture and ASK: Is this angle co-interior with the angle given?
Have students signal the answer with thumbs up or thumbs down. When students have
identified the correct angle as co-interior, ASK: How can we find its measure? To prompt
students to see the answer, label the angles as shown below and ask them to find angle x
before finding angle y. (the lines are parallel, so the corresponding angles are equal, so
x = 138; angles x and y are supplementary, so y = 180 − 138 = 42)
138
x
y
ASK: Do you think this will work for any pair of parallel lines that have a third line intersecting
them? (yes) What do co-interior angles add to when lines are parallel? (180) In other words,
what can we call co-interior angles at parallel lines? (supplementary angles) Write on the board:
When lines are parallel, co-interior angles are supplementary so they add to 180.
Exercises: Find the missing angle measures.
a)
b)
Bonus:
y
x
y
z
52
121
b
70
x
a
Answers: a) x = 128, y = 52; b) a = 121, b = 59, Bonus: x = y = 130, z = 70
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-97
Extensions
(MP.3) 1. SAY: Let’s see what happens if we have two lines that intersect. Can we still have
equal corresponding angles? Draw on the board:
x
y
x
SAY: It looks like these two lines have equal corresponding angles. Maybe I did not draw them
perfectly, but I’ve marked the angles I think are equal with an x. Circle the x at the bottom and
SAY: I see a triangle in this picture. ASK: What do we call the angle I circled for that triangle?
(external angle) What do we know about the measure of the external angle in a triangle?
(the measure of the external angle equals the sum of the measures of the interior angles
opposite it) How can we write it down using the letters in the diagram? (x = x + y) What does
this say about the measure of angle y? ( y = 0) Can that happen? (no) SAY: If the lines have
equal corresponding angles they cannot intersect, because if they do we get a 0 angle between
them. In mathematics, when you suppose something happens and you logically arrive at
nonsense, you conclude that your original idea was incorrect. So in this case, we proved using
logic that, if two lines meet a third line making equal corresponding angles, the lines are parallel.
(MP.3) 2. Use co-interior angles to explain why opposite angles in a parallelogram are equal.
Answer:
B
A
C
D
AB || CD and angles B and C are co-interior angles, so C = 180 − B.
AD || BC and angles B and A are co-interior angles, so A = 180 − B.
This means C = A.
Similarly, AB || CD and angles B and C are co-interior angles, so B = 180 − C.
AD || BC and angles C and D are co-interior angles, so D = 180 − C.
This means B = D.
3. Demonstrate the equality between corresponding angles using translation—a slide. Make two
copies of the picture below on transparencies. Place the transparencies one on top of the other
and show students that they are identical. Then slide one transparency down the other so that
the line KQ slides along the line KL to the position of LR, so that MKQ slides to match MLR.
This means MKQ = MLR.
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Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Have students complete the statements below.
a) KQ is parallel to ____ and RLK and QKM are _______________ angles,
so QKM = RLK = 70°.
b) QKM and TQS are corresponding angles and QKM = 70° = TQS, so ____ is
parallel to _____.
c) The quadrilateral QRLK is a ____________________.
Answers: a) LR, corresponding; b) KL, RS; c) parallelogram
(MP.1, MP.3, MP.7) 4. Draw a parallelogram. Measure each angle of the parallelogram and
draw a ray that divides each angle into two equal angles. Extend each ray so that it intersects
two other rays. What geometric shape can you see in the middle of the parallelogram? Use the
sum of the angles in the shaded triangle to explain why this is so.
Answer: The shape in the middle is a rectangle. The acute angles of the triangle are both half
of the angles of a parallelogram. The adjacent angles of a parallelogram add to 180, so their
halves add to 90. Because the sum of the angles in a triangle is 180, the third angle in each
triangle is a right angle. Since there are three other triangles like the shaded triangle in the
parallelogram, the shape in the middle has four right angles and must be a rectangle.
(MP.1, MP.3) 5. A restaurant has many windows of unusual shapes. One of them is the
trapezoid shown below.
A
60
7 cm
120
10 cm
D
5 cm
114
66
B
6.2 cm
C
The restaurant owner, Bill, calls a window company to order a replacement for this window. He
tells the company representative that the window is a trapezoid with AB parallel to CD, and he
gives some more information.
For each combination of information below, say whether Bill is giving enough information to
create a unique trapezoid of the exact shape and size wanted. If so, provide directions for
constructing the trapezoid if they are easy. If there is no unique trapezoid, draw two trapezoids
with the given specifications. Hint: Draw a line parallel to AD through point C. It separates the
trapezoid into a quadrilateral and a triangle. Label the new line EC and label the triangle EBC.
What type of quadrilateral have you created? Can you construct this quadrilateral and the
leftover triangle given the additional information?
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-99
a) window is a trapezoid with AB parallel to CD; angles A and B and sides AB and DA
b) window is a trapezoid with AB parallel to CD; angles A and B and sides AB and CD
c) window is a trapezoid with AB parallel to CD; angles A and B and sides BC and DA
d) window is a trapezoid with AB parallel to CD; sides AB, BC, CD and angle B
e) window is a trapezoid with AB parallel to CD; sides AB, BC, CD, DA and angle A
f) window is a trapezoid with AB parallel to CD; the four sides, AB, BC, CD, DA
Answers:
a) Yes, the information Bill gives creates a unique trapezoid. Construct line AB and two rays
to form angles A and B, and then draw the side DA = 7 cm. Draw a line parallel to AB through
D and label it DC. The ray creating angle B intersects with line DC at C and finishes the
unique trapezoid.
b) Yes. To create the combined shape, create the triangle before the quadrilateral
(a parallelogram), but determine the triangle sides and angles from the parallelogram.
The parallelogram AECD will have equal opposite sides, so CD = AE = 5 cm and AD = EC.
Since AD || EC, A and CEB are corresponding angles at parallel lines and they are equal
(60 each). Thus, in triangle CEB, CEB = 60, CBE = B = 66, and side EB = 5 cm
(because AB = 10 cm and AE = DC = 5 cm, AB − AE = EB, thus 10 − 5 = 5 cm). Start by
constructing the triangle. Then extend side BE beyond point E 5 cm to point A. Complete the
parallelogram by drawing angle A = 60 for line AD and drawing a line CD parallel to AB
through point C.
c) No. Without knowing the length of side AB, we don’t know where to construct angle B, so this
leaves both sides AB and CD of unknown lengths.
A
60
7 cm
120
x cm
D
y cm
114
66
B
6.2 cm
C
d) Yes. Construct the side AB and angle B. Construct the side BC. Since CD  AB,
C = 180 − B = 114, so construct angle C so that CD is 5 cm. Join points A and D to
create the fourth side.
e) Yes. Construct line AB, angle A, and line AD. Since ABCD is a trapezoid, D is a co-interior
angle with A, so D = 180 − A = 120. Construct D and side CD (5 cm). Join points B
and C.
f) Yes. To create the combined shape, create the triangle before the parallelogram, but
determine the triangle sides from the parallelogram. The parallelogram AECD will have equal
opposite sides, so AE = DC and AD = EC. Start the triangle with the sides EB = 5 cm (AB − EB
= AE = DC = 5 cm), BC = 6.2 cm, and EC = 7 cm (AD = 7 cm = EC). Extend side BE beyond
point E 5 cm to point A. From point C, extend line CD 5 cm parallel to line AB and join AD
(7 cm) to complete the parallelogram.
D-100
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-15
Alternate Angles and Parallel Lines
Pages 109–111
Standards: 8.G.A.5
Goals:
Students will use informal arguments to establish facts about alternate angles at parallel lines.
They will use these facts to informally prove that angles in a triangle add to 180º.
Prior Knowledge Required:
Can identify parallel lines
Can identify supplementary, vertical, corresponding, and co-interior angles
Knows the properties of supplementary, vertical, corresponding, and co-interior angles
Can draw and measure with a ruler and a protractor
Knows what counterexamples are
Vocabulary: alternate angles, co-interior angles, conjecture, corresponding angles,
counterexample, parallel, straight angle, supplementary angles, vertical angles
Materials:
rulers
transparencies
overhead projector
protractors
The Geometer’s Sketchpad®
Introduce alternate angles. Draw on the board:
1 2
4
3
5
7
6
8
Explain that alternate angles are angles that create a pattern like in the letter Z. For example,
angles 3 and 6 in the picture are alternate angles. Trace the letter Z on the picture with your
finger. Copy the initial picture, trace the letter Z, and have students find the other pair of
corresponding angles, as shown at right below:
1
3
5
7
8
6
2
3
4
5
7
1 2
4
6
8
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-101
If students do not see that the reflected pattern resembles a letter Z, copy the picture onto a
transparency, highlight the letter Z, and turn it over so that students see the pattern.
Exercises: List the alternate angles.
a)
5
1
6
2
b)
3 4
7 8
a
c
b
d
k
m
n
u
Answers: a) 6 and 3, 2 and 7; b) d and k, b and n
Alternate angles for parallel lines are equal. Have students draw a pair of parallel lines by
using the opposite sides of a ruler. Ask them to place the ruler across the lines they drew and
draw a third line intersecting both the first and second lines. Have them identify both pairs of
alternate angles and measure them. ASK: What do you notice about the measures of alternate
angles? (they are equal) Did everyone create angles of the same size? (no) Did everyone get
the same result: alternate angles in a pair of parallel lines are equal? (yes)
ASK: Are alternate angles always equal? Have students repeat the exercise above, this time
starting with a pair of lines that are not parallel.
Students can also do Activity 1 below to discover that parallel lines create equal alternate
angles. Part e) encourages students to pay close attention to what they see and to what
changes on the screen.
Activity 1
Use The Geometer’s Sketchpad® for this activity.
Discovering that parallel lines create equal alternate angles.
a) Draw a line. Label it AB. Mark a point C not on the line, and construct a line parallel to AB
through C.
b) Draw a line through points B and C.
c) On the line parallel to AB, mark point D on the other side of BC from point A.
d) Name two alternate angles in the diagram.
e) Measure the alternate angles you named. What do you notice? Make a conjecture.
f) Move the lines or the points A, B, or C around. Are the alternate angles always equal?
(MP.5) g) Pull point D to the other side of the line BC to stop the angles you measured being
equal. Look at the pattern the angles create. Are they alternate angles? Does this create a
counterexample to your conjecture from e)? Explain.
Answers: e) alternate angles at parallel lines are equal; f) yes; g) When angles stop being
equal, they are not alternate angles anymore. This does not create a counterexample because
the statement talks about alternate angles and the unequal angles are not alternate.
(end of activity)
D-102
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Proving that alternate angles at parallel lines are equal. To review supplementary, vertical,
corresponding, and co-interior angles, draw the picture below on the board and have students
find all the missing angle measures. Ask them to explain how they found each angle measure.
Encourage multiple explanations. (sample answers: b is co-interior with the given angle, so its
measure is 180 − 40 = 140; a and the given angle are corresponding angles, so they are
equal; angles a and b are supplementary angles, so b = 180 − 40 = 140)
a
b
c
40
d
m
n u
ASK: Which angles in this picture are alternate angles? (d and the given angle, b and n)
Are they equal? (yes) Are alternate angles at parallel lines always equal? (yes) How can we
explain using logic that these angles are equal? Replace the angle measure with the letter k and
have students explain why angles k and d are equal. PROMPT: Which angle is
corresponding with k? (a) What do we know about corresponding angles? (they are equal)
What do we know about angles a and d? (they are equal) Why? (they are vertical angles)
Find measures of angles between intersecting lines using alternate, corresponding,
vertical, and supplementary angles.
(MP.7) Exercises: Find all the angles in the picture.
a)
b)
54
62.4
b
c)
a
119.77
c
Bonus:
61.23
d
119.77
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
55°
65°
D-103
Answers:
a)
54 126 54
b)
126 54 126
54 126 54
90°
62.4 117.6 90 90
117.6 62.4
62.4 117.6
c)
b
119.77
c
119.77
60.23
a
60.23
60.23
119.77
Bonus:
118.77
65°
60.23
61.23
61.23 118.77
119.77
d
90 90
90 90
120°
125° 55°
55°
60°
120°
60°
65°
125°
Using alternate angles to identify parallel lines. Remind students that when they learned
about corresponding angles, they looked at two statements: When lines are parallel,
corresponding angles are equal, and when corresponding angles are equal, lines are parallel.
Write on the board:
When lines are parallel, alternate angles are equal.
ASK: Is this true or false? (true; we have proved it using logic) Write “true” beside the statement.
ASK: If we change the order in this statement, what statement will we get? (when alternate
angles are equal, the lines are parallel) Write that statement on the board and explain that you
want to investigate whether this statement is true. ASK: What do you need to do? (draw a pair
of lines with equal alternate angles and check whether they are parallel) How can we check if
lines are parallel? (For example, if the corresponding angles are equal, then the lines are parallel)
Draw on the board:
x
x
(MP.3) SAY: These lines have equal alternate angles. Which other angles do we know are
equal to x? (vertical angles to the given ones) Have a volunteer mark the angles and explain
why they are equal to x. ASK: What other types of angles can we see in this picture?
(corresponding angles) Have another volunteer circle a pair of corresponding angles.
ASK: Are the corresponding angles equal? (yes) What do we know about lines that have equal
corresponding angles? (they are parallel) SAY: We have just proved that these two lines are
parallel, so we have just proved that the statement is true. Write “true” beside the second
statement on the board.
D-104
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Practice identifying parallel lines using the equality between alternate angles. Return to
the picture in part c) in the previous exercises and ask students to circle a pair of corresponding
angles for each pair of lines, using different colors. Ask students to identify which rays are
parallel. (rays b and c)
Activity 2
Have students work in pairs to complete this activity.
a) Draw two pairs of lines, one pair that is parallel and the other not but looking like it might be.
(For example, draw a line and then place a ruler along the line as if to draw another line along
the parallel side of the ruler, but then rotate the ruler very slightly.) Do not indicate which pair
is which.
b) For each pair of lines, draw a third line intersecting the first and second line.
c) Exchange your paper with a partner. Measure and compare the corresponding angles to
identify the pair of parallel lines.
(end of activity)
(MP.3) Proving that sum of the angles in a triangle is 180. Explain that the properties of
alternate angles allow us to prove using logic that the sum of the angles in a triangle equals
180. Point out that, even though we discovered this fact and used it many times, we have not
proven it using logic. Draw a triangle on the board and label the angles a, b, and c as shown
below. Explain that you want to use alternate angles, so you are drawing a line parallel to one of
the sides, through the opposite vertex. Draw the line, as shown below:
b
a
c
ASK: Are there any equal alternate angles in this picture? (yes) Is there an angle alternate to
angle a? (yes) Point to different angles and have students signal thumbs up or thumbs down to
show if this is the angle alternate with a. Repeat with c. Point to the three angles at the top of
the picture and ASK: What type of angle do the three angles make? (straight angle) What is the
measure of a straight angle? (180) Ask students to write an equation that shows that the three
angles add to 180. (a + b + c = 180) ASK: What did we need to prove to show that angles in a
triangle add to 180? (a + b + c = 180) SAY: We have just proven precisely that fact.
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-105
Extensions
1. Demonstrate the equality between corresponding angles using transformations. Make two
copies of the picture below on transparencies. Place the transparencies one on top of the other,
and show students that they are identical. Have students identify a pair of alternate angles
(e.g., QKL and NLK). Mark the angles with arcs.
Q
O
a) Slide the line KQ down along the ray KL to the position of LR so that MKQ slides to MLR.
This means MKQ = MLR. The angles MLR and NLK are vertical angles, so they are
equal. But MLR = MKQ, so MKQ = NLK.
b) Press a pencil to act as a pivot to the point O. Rotate the top transparency 180 around
point O to show how angle QKL becomes NLK.
(MP.3) 2. Sketch parallelogram ABCD.
A
D
B
C
Fill in the blanks to prove using logic that opposite sides of a parallelogram are equal.
a) Parallel lines AB and _____ intersect with the third line DB. ABD and ____ are alternate
angles at parallel lines, so ABD = _____. Label these angles with the same number of arcs.
b) Parallel lines AD and _____ intersect with the third line DB. ADB and ____ are alternate
angles at parallel lines, so ADB = _____. Label these angles with the same number of arcs.
c) Triangles ABD and ______ are congruent by the _____ congruence rule because they have
two pairs of equal corresponding _____________ and a common ______________.
d) ABD  ______, so AB = ____ and AD = ______.
Answers: a) CD, CDB, CDB; b) BC, CBD, CBD; c) CDB, ASA, angles, side; d) CDB, CD, CB
D-106
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
G8-16
Solving Problems Using Angle Properties
Pages 112–113
Standards: 8.G.A.2, 8.G.A.5
Goals:
Review of the material learned in the unit.
Prior Knowledge Required:
Can identify parallel lines
Can identify supplementary, vertical, co-interior, corresponding, and alternate angles
Knows the properties of supplementary, vertical, co-interior, corresponding, and
alternate angles
Can draw and measure with a ruler and a protractor
Knows what counterexamples are
Knows the properties of triangles and special quadrilaterals
Can identify congruent triangles
Can apply the congruence rules for triangles
Vocabulary: alternate angles, co-interior angles, congruent, corresponding angles,
counterexample, diagonal, equilateral, isosceles trapezoid, isosceles triangle, kite, parallel,
right trapezoid, right triangle, scalene, supplementary angles, vertical angles
Materials:
scissors
BLM Triangles for Making Quadrilaterals (p. D-132)
BLM Angle Properties (Summary) (p. D-133, optional)
BLM Properties of Angles in a Triangle (Summary) (p. D-134, optional)
This lesson serves as a review and combination of all the material learned in this unit.
The exercises can be assigned or you can work through them as a class to review the
angle properties.
Exercises: Find the missing angles.
a)
A
B
b)
c)
37°
x
x
84°
z
D
y
w
x
u
C
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
a
50°
y
z
y
46°
b
72° z
D-107
d)
e)
34°
F
e
E
d
z
d
d 57
G c
w
f
H
x
y
54°
u
20°
v
Answers: a) x = 53, y = 37, z = 53; b) u = 134, w = 46, x = 84, y = 130,
z = 50; c) a = x = 62, b = 118, y = 44; d) c = f = 63, d = 60, e = 120;
e) w = y = 126, x = 20, z = 34, u = v = 63
Have students explain how they found the angles in the previous exercises. Then discuss what
other additional information, such as types of quadrilaterals, parallel lines, or congruent triangles
we can deduce from the diagrams in the exercises. For example, in part a) lines AB and CD are
parallel because the co-interior angles ABC and BCA, both right angles, add to 180, so ABCD
is a trapezoid. In part d) triangle EFG is an equilateral triangle because all of its angles are
equal, but triangle FGH is an acute scalene triangle because it has no equal angles. We can
also deduce that lines EF and GH are not parallel. In part e) we have two pairs of congruent
triangles by the angle-side-angle (ASA) congruence rule, so we can mark other equal line
segments. One pair of triangles has angles 34, 20, and 126 and a common vertex, and the
other pair has a common side and angles 34, 83, and 63.
Write and draw on the board:
A quadrilateral with two
pairs of equal adjacent
sides is called a kite.
A trapezoid with equal
opposite sides is called an
isosceles trapezoid.
A trapezoid with two
right angles is called a
right trapezoid.
NOTE: The activity below helps students produce counterexamples to various statements about
quadrilaterals. Keep the above definitions on the board to help students during the activity.
Make sure students know that a line connecting the opposite vertices of a quadrilateral is called
a diagonal.
(MP.3) Activity
Cut out the triangles from BLM Triangles for Making Quadrilaterals. Check the white triangles
for sides and angles equal to sides or angles of the gray triangle. Mark the equal sides and
angles on all triangles, on both sides of each triangle.
D-108
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
Combine the gray triangle with one of the other triangles in different ways so that they share a
side—in other words, two sides line up to form the diagonal for a new shape.
Use the triangles you cut out to answer the questions.
a) Which triangle combined with the gray triangle allows you to make a parallelogram? Use
equal angles to explain why the shape you produced is a parallelogram. Make more than one
parallelogram from this pair of triangles. Make a quadrilateral that is not a parallelogram and
explain how you know it is not a parallelogram.
b) Use two different right triangles to produce a quadrilateral that has exactly one pair of equal
opposite angles and that is not a kite.
c) Find all triangles that allow you to make a trapezoid with the gray triangle. Explain why the
shape you produced is a trapezoid. What kind of trapezoid is it?
d) Combine the two triangles you picked in part c). Can you make a trapezoid from these two
triangles? Explain your answer. Is it the same type of trapezoid as in part c)?
Answers:
a) A, you can make three different parallelograms (one of them a rectangle) and a kite. Sample
explanation for parallelogram:
a
b
b
a
Triangle A is congruent to the gray triangle, and both are right triangles. This means that the
acute angles in both triangles add to 90, or b = 90 − a. The adjacent angles in the
quadrilateral are then a and b + 90 = 90 + (90 − a) = 180 − a. The adjacent angles
add to 180 and they are co-interior angles for the opposite sides of the quadrilateral. This
means the opposite sides of the quadrilateral are parallel. The kite is not a parallelogram
because parallelograms have equal opposite sides, and the kite has opposite sides that are
not equal.
c) Triangles E and F allow you to make a right trapezoid with the gray triangle. They both have
at least one angle that is the same as the smallest angle of the gray triangle.
Sample explanation:
b
a
a
a
The angles marked with an arc are equal, and they are alternate angles for the top and the
bottom side of the quadrilateral. When alternate angles are equal, the lines are parallel, so the
top and the bottom sides of the quadrilateral are parallel. The angles a and b adjacent to the two
angles just discussed are alternate angles for the other two sides and they are not equal, so
these sides are not parallel. The quadrilateral is a trapezoid. We also know that a + b = 90, so
the top left angle of the trapezoid is a right angle. So the trapezoid has two right angles and is a
right trapezoid.
d) Triangles E and F together can produce a trapezoid, but it is not a right trapezoid. The
explanation is similar to the one in part c).
(end of activity)
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry
D-109
You might wish to provide students with BLM Angle Properties (Summary) and BLM
Properties of the Angles in a Triangle (Summary) as summaries of material learned in
this unit.
Extensions
(MP.3) 1. a) What is the sum of the angles in a quadrilateral?
b) Draw a quadrilateral with exactly three equal angles (see two samples below). Is it a special
quadrilateral?
c) Can a quadrilateral with exactly three equal angles be a parallelogram? A trapezoid? Use
what you know about the sum of adjacent angles in a parallelogram or a trapezoid to explain
your answer.
d) A special quadrilateral has exactly three equal angles. What type of special quadrilateral is it?
Answers:
a) 360
b) Answers will vary, most likely no.
c) No for both. Explanation: Suppose the shape is a trapezoid. Angles adjacent to a non-base
side of a trapezoid add to 180° because they are co-interior angles at parallel sides. At least two
of the three equal angles have to be adjacent to the same non-base side, so they have to add to
180 and so have to be 90°. If the three equal angles are all right angles, the fourth angle is
360° − 3 × 90° = 90°. This shape has four equal angles and not three, so a quadrilateral with
exactly three equal angles cannot be a trapezoid. The explanation for a parallelogram is very
similar.
d) The only type of special quadrilateral that is not a trapezoid or a parallelogram is a kite.
(MP.3) 2. a) Draw a line AC and mark a point B on it. Draw a line segment BD intersecting AC.
b) Using a protractor, draw rays BE and BF so that ray BE divides ABD into two equal angles
and ray BF divides DBC into two equal angles as shown below:
c) Find the measure of EBF without using a protractor. Explain your answer. Verify your
answer using a protractor.
Answer: c) EBF = 90. Explanation: ABE = EBD, DBF = FBC,
and ABE + EBD + DBF + FBC = 180
However, ABE + EBD + DBF + FBC = EBD + EBD + DBF + DBF
= (EBD + DBF) + (EBD + DBF)
= EBF + EBF
So EBF = 180 ÷ 2 = 90.
D-110
Teacher’s Guide for AP Book 8.1 — Unit 3 Geometry