Download Teacher`s Guide 7.1

G7-3 M
easuring and Drawing Angles
and Triangles
Pages 102–104
Goals
Curriculum
Expectations
Students will measure and draw angles.
Ontario: 5m51, 5m52, 5m54, 6m48, 6m49, 7m3, 7m4,
7m46
WNCP: 6SS1, review,
[T, R, V]
PRIOR KNOWLEDGE
REQUIRED
Materials
Knows what an angle is
Can name an angle and identify a named angle
dice
geoboards and elastics
grid paper
Dynamic geometry software
(optional)
Vocabulary
angle
vertex
arms
acute
obtuse
Review the concept of an angle’s size. Draw two angles:
ASK: Which angle is smaller? Which corner is sharper? The diagram on the
left is larger, but its corner is sharper, and mathematicians say that this angle
is smaller. The distance between the ends of the arms in both diagrams
is the same, but this does not matter; angles are made of rays and these
can be extended. What matters is the “sharpness.” The sharper the corner
on the “outside” of the angle, the narrower the space between the angle’s
arms. Explain that the size of an angle is the amount of rotation between the
angle’s arms. The smallest angle is closed; both arms are together. Draw the
following picture to illustrate what you mean by smaller and larger angles.
Smaller
Larger
You might also illustrate what the size of an angle means by opening a book
to different angles. Draw some angles and ask your students to order them
from smallest to largest.
A
E-6
B
C
D
E
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
You can show how much an angle’s arm rotates with a piece of chalk. Draw
a line on the board then rest the chalk along the line’s length. Fix the chalk
to one of the line’s endpoints and rotate the free end around the endpoint to
any desired position.
Define acute and obtuse angles in relation to right angles. Obtuse angles
are larger than a right angle; acute angles are smaller than a right angle.
ExTRa PRaCTICE:
1. Copy the shapes onto grid paper and mark any right, acute, and
obtuse angles. Which shape has one internal right angle? What did
you use to check?
2. Which figures at left have
a) all acute angles?
b) all obtuse angles?
c) some acute and some obtuse angles?
Introduce protractors. On the board, draw two angles that are close to
each other—say, 50° and 55°—without writing the measurements and in
a way that makes it impossible to compare the angles visually. aSK: How
can you tell which angle is larger? Invite volunteers to try different strategies
they suggest (such as copying one of the angles onto tracing paper and
comparing the tracing to the other angle, or creating a copy of the angle
by folding paper). Lead students to the idea of using a measurement tool.
ExaMPLE:
Explain that to measure an angle, people use a protractor. A protractor
has 180 subdivisions around its curved edge. These subdivisions are
called degrees ( °). Degrees are a unit of measurement, so just as we write
cm or m when writing a measurement for length, it is important to write
the ° symbol for angles.
origin
base line
Using protractors. Show your students how to use a protractor on the
board or on the overhead projector. Identify the origin (the point at which
all the degree lines meet) and the base line (the line that goes through
the origin and is parallel to the straight edge). When using a protractor,
students must
• place the vertex of the angle at the origin;
• position the base line along one of the arms of the angle.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
You could draw pictures (see below) to illustrate incorrect protractor use,
or demonstrate it using an overhead projector and a transparent protractor.
Geometry 7-3
E-7
Introduce the degree measures for right angles (90°), acute angles (less than
90°), and obtuse angles (between 90° and 180°). Point out that there are
two scales on a protractor because the amount of rotation can be measured
clockwise or counter-clockwise. Students should practise choosing the
correct scale by deciding whether the angle is acute or obtuse, then saying
whether the measurement should be more or less than 90°. (You may want
to do some examples as a class first.)
Then have students practise measuring angles with protractors. Include
some cases where the arms of the angles have to be extended first.
Introduce angles in polygons, then have students measure the angles in
several polygons. Students can draw polygons (with both obtuse and acute
angles) and have partners measure the angles in the polygons.
Drawing angles. Model drawing angles step by step (see Workbook p.
104 or BLM Measuring and Drawing Angles and Triangles, p E-42).
Emphasize the correct position of the protractor. Have students practise
drawing angles. You could also use Activity 2 for that purpose.
Have students practise drawing lines that intersect at a given angle.
Another way to practise drawing angles is to construct triangles with given
angle measures.
ACTIVITIES 1–4
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 different numbers of given angles. EXAMPLES:
d)
e)
12°
9°
4°
process Expectation
Selecting tools and
strategies, Technology
E-8
2. Students will need a die, a protractor, and a sheet of paper. Draw a
starting line on the paper. Roll the die and draw an angle of the measure
given by the die; use the starting line as your base line and draw the
angle counter-clockwise. Label your angle with its degree measure.
For each next roll, draw an angle in the counter-clockwise direction so
that the base line of your angle is the arm drawn at the previous roll.
The measure of the new angle is the sum of the result of the die and the
measure of the angle in the previous roll. Stop when there is no room to
draw an angle of the size given by the roll. For example, if the first three
rolls are 4, 5, and 3, the picture will be as shown.
3. Teach students to draw and measure angles using Geometer’s
Sketchpad. Then ask them to try moving different points (on the arms
of the angle, or its vertex) so that the size of the angle becomes, say,
50°. Is it easy or hard to do? When you move the line segments, does
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
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
Sample ANSWERS:
the angle change? (yes) When you move the vertex or other point on
the arms, does the angle change? (yes) Show students how to draw
an angle of fixed measure (using menu options). Will moving the
endpoints change the size of the angle now? (no) Show students how
to draw angles equal to a given angle.
4. Have students draw polygons in Geometer’s Sketchpad and
measure the size of the angles and the length of the sides of these
polygons. Have students check that the angle measures they obtain
make sense. For example, clicking on three vertices of a quadrilateral
and then using menu options to measure the angle might produce
different angles, depending on the order in which the vertices were
selected. Also, the software sometimes measures angles in the wrong
direction, producing an answer more than 180°.
process Expectation
Technology, Reflecting
on the reasonableness
of the answer
Extensions
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
1. Angles on an analogue clock.
What is the angle between the hands at 12:24? 13:36? 15:48? (Draw the
hands first!) To guide students to the answers, draw an analogue clock
that shows 3:00 on the board. Ask your students what angle the hands
create. What is the measure of that angle? If the time is 1:00, what is the
measure of the angle between the hands? Do you need a protractor to
tell? Have students write the angle measures for each hour from 1:00 to
6:00. Which number do they skip count by?
An hour is 60 minutes and a whole circle is 360°. What angle does the
minute hand cover every minute? (6°) How long does it take the hour
hand to cover that many degrees? How do you know? (12 minutes,
because the hour hand covers only one twelfth of the full circle in an
hour, moving 12 times slower than the minute hand)
If the time is 12:12, where do the hour hand and the minute hand point?
What angle does each hand make with a vertical line? What is the angle
between the hands? (ANSWER: The hour hand points at one minute
and the angle that it makes with the vertical line is 6°. The minute hand
points at 12 minutes and the angle that it makes with the vertical line is
12 × 6 = 72°. The angle between the hands is 72° − 6° = 64°.)
2. Some scientists think that moths travel at a 30° angle to the sun when
they leave home at sunrise. Note that the sun is far away, so all the rays
it sends to us seem parallel.
sun’s rays
flower
N
30°
evening
Geometry 7-3
moth home
W
morning
E
S
E-9
a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the sun in the evening?
b) A moth sees the light from the candle flame and thinks it’s the sun. The candle is very near to us, and the rays it sends to us go out in all directions. Where does the moth end up? Draw the moth’s path.
30°
30°
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
E-10
Teacher’s Guide for Workbook 7.1
G7-4 Perpendicular Lines
Pages 105–106
Goals
Curriculum
Expectations
Students will identify and draw perpendicular lines.
Ontario: 7m4, 7m5, 7m46
WNCP: 6SS1, 7SS3,
[CN, V, T]
PRIOR KNOWLEDGE REQUIRED
Materials
Knows what a right angle is
Can name an angle and identify a
named angle
Dynamic geometry software
(optional)
protractors
set squares
Vocabulary
angle
perpendicular
slant line
right angle
arms
Introduce perpendicular lines (lines that meet at 90°) and show how
to mark perpendicular lines with a square corner. Draw several pairs of
intersecting lines on the board and have students identify the perpendicular
lines. Include pairs of lines that are not horizontal and line segments that
intersect in different places and at different angles (see examples below).
Invite volunteers to check whether the lines are perpendicular using a corner
of a page, a protractor, and/or a set square.
Ask students where they see perpendicular lines or line segments—also
called perpendiculars—in the environment (sides of windows and desks,
intersections of streets, etc.).
Extra practice:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Which lines look like they are perpendicular?
K
C
P
B
F
a)
b)
c) J
d) O
L
G
Q
D
A
H
M
I
R
N
E
A perpendicular through a point. Explain that sometimes we are interested
in a line that is perpendicular to a given line, but we need an additional
condition—the perpendicular should pass through a given point. In each
diagram below, have students identify first the lines that are perpendicular to
the segment AB, then the lines that pass through point P, and finally the one
line that satisfies both conditions.
G
D
K
a)
b)
E
G
B
P
C
F
P
A
C
B
F
A
D
H
E
H
Constructing a perpendicular through a point. Model using a set square
(and then a protractor) to construct a perpendicular through a point that
is not on the line. Emphasize the correct position of the set square (one
Geometry 7-4
E-11
side coinciding with the given line, the other touching the given point) and
the protractor (the given line should pass through the origin and through
the 90° mark). Have students practise the construction. Circulate among
the students to ensure that they are using the tools correctly. Then invite
volunteers to model the construction of a perpendicular through a point that
is on the line. (Emphasize the difference in the position of the set square:
the square corner is now at the point, though one arm still coincides with
the given line.) Then have students practise this construction as well.
Extra practice:
Draw a pair of perpendicular slant lines (i.e., lines that 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? (rectangle)
Bonus
Draw a slant line and a point not on the line. Using a protractor
and a ruler or set square, 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 point to the given line. Measure the distance from
the point to the line along the perpendicular. Then mark a point on the given
line that is the same distance from the intersection as the given point. Now
draw a perpendicular to the given line through this point as well. Finally,
draw a perpendicular to the last line through the given point.)
connection
Science
Why perpendiculars are important. Discuss with students why
perpendiculars are important and where are they used in real life. For
example, you can explain that it is easy to determine a vertical line (using
gravity—just hang a stone on a rope and trace the rope), but you need
a right angle to make sure that the floor of a room is horizontal.
process Expectation
1. Have students use Geometer’s Sketchpad to:
Technology
a) Draw a line. Label it m.
b) Mark a point A on the line m.
c) Draw another line through point A.
d) Measure the angle between the two lines.
e) Try to make the angle a right angle by moving the points
around. Is it easy or hard to do?
f) Check whether the lines stay perpendicular when you move any
of the points in the picture.
g) Repeat parts a) through f) with a point not on the line. Note
that when point A is not on the line, the second line might be
modified so that it does not intersect the line m, and the angle
you measure disappears.
Explain that you need a method to draw perpendicular lines that
will keep them perpendicular even if the points are moved around.
E-12
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
ACTIVITIES 1–2
Teach students to draw perpendicular lines using the perpendicular
line command from the menu. Do these lines stay perpendicular
to the given line even if points are moved around? (yes)
2. Have students draw a triangle using the polygon tool in
Geometer’s Sketchpad. Ask them to move the points around to
make it look like a right triangle. Then ask them to measure the
angles of the triangle and to check whether it is indeed a right
triangle. 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) Ask students to think about how they draw a right
triangle on paper. What tools do they use and why? (a protractor
or a set square) What tools could we use instead of protractors
and set square in Geometer’s Sketchpad? (perpendicular lines)
Then have students draw a right triangle in Geometer’s Sketchpad.
Ask them to check that the triangle remains a right triangle even if
points are moved around.
process Expectation
Technology
process Expectation
Selecting tools
and strategies
Students can also add the measures of the acute angles in the
right triangle they created, and check that the sum remains 90°
even when the triangle is modified.
Extension
process Expectation
Visualizing
Discuss with your 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.
P
P
B
A
B
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
Geometry 7-4
E-13
G7-5 Perpendicular Bisectors
Pages 107–108
Goals
Curriculum
Expectations
Students will identify and draw perpendicular bisectors.
Ontario: 7m1, 7m3, 7m4,
7m48
WNCP: 6SS1, 7SS3,
[C, R, V, T]
PRIOR KNOWLEDGE REQUIRED
Can identify and construct a perpendicular using a set square
or a protractor
Can identify and mark right angles
Can name line segments and identify named line segments
Can draw and measure with a ruler
Vocabulary
line segment
midpoint
right angle
perpendicular
bisector
Example:
Materials
paper circles or BLM Circles (p E-51)
Dynamic geometry software
(optional)
Introduce the notation for equal line segments. Explain that when we
want to show that line segments are equal, we add the same number of
marks across each line segment. This is particularly useful for sketches,
when you are not drawing everything exactly to scale.
Introduce the midpoint. Model finding the midpoint of a segment using a
ruler, and mark the halves of the line segment as equal, then have students
practise this skill.
Introduce bisectors. Explain that a bisector of a line segment is a line
(or ray or line segment) that divides the line segment into two equal parts.
There can be many bisectors of a line segment. Ask students to draw a
line segment with several bisectors.
ASK: Can a line have a bisector? What about a ray? (no, because lines
and rays have no midpoints)
Draw a scalene triangle. Choose a side and draw a bisector to that side
that passes through the vertex of the triangle that is opposite that side.
Repeat with the other sides. What do you notice? (ANSWER: All three
bisectors, called medians, pass through the same point.)
Introduce perpendicular bisectors. Of all the bisectors of a line segment
only one is perpendicular to the line segment, and it is called the
perpendicular bisector. The perpendicular bisector of a line segment
• divides the line segment into two equal parts AND
• intersects the line segment at right angles (90°).
Point out that there are two parts in the definition, and both must be true.
ASK: How can we draw a perpendicular bisector? How is that problem
E-14
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Extra practice:
similar to constructing a bisector? Constructing a perpendicular? Lead
students to the idea that they should first find the midpoint of the line
segment, then construct a perpendicular through that point. Have students
practise drawing perpendicular bisectors using set squares and protractors.
process Expectation
Changing into a
known problem
E
D
F
J
G
C
H
B
Present several diagrams that combine intersecting line segments,
perpendicular lines, and bisectors, such as the one at left, and have students
identify equal segments, then find perpendicular lines and bisectors.
(Example: Find a bisector of EG. Is it a perpendicular bisector?) Ask other
questions about the diagram, such as:
• J is the midpoint of what segment? (CG) Why not AD? How do you
know? (We do not have any information about the lengths of DJ and
AJ. They look equal, but might have different lengths.)
• Name three line segments GJ is perpendicular to.
A
Finally, have students identify all the perpendicular bisectors and the line
segments they bisect. (ANSWERS: AD, JA, and JD bisect line segment CG;
CJ, JG, and CG bisect line segment FH)
process assessment
Workbook Question 8f) –
[C,], 7m3
Ask students to find examples of equal segments, midpoints, and
perpendicular bisectors in the classroom or elsewhere, such as in letters
of the alphabet, in pictures or photographs, and so on.
ACTIVITIES 1–3
process Expectation
Technology, Selecting
tools and strategies
2. Paper folding and line segments
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
A
C
B
1. Show students how to construct the midpoint of a line segment
in Geometer’s Sketchpad. Explain that there is no command that
will construct a perpendicular bisector of a line segment. Challenge
students to find a way to construct a perpendicular bisector of a
line segment using the tools they are familiar with in Geometer’s
Sketchpad. (Find the midpoint of the line segment, then construct a
line through the midpoint and perpendicular to the given line segment)
PROMPT: Think of the way you construct a perpendicular line segment
on paper. Which tools can you use? (protractor or set square and
ruler) Which tool replaces a protractor or a set square in Geometer’s
Sketchpad? (“Perpendicular Line” in the “Construct” menu)
Draw a line segment AB dark enough that you can see through the
paper. Fold the paper so that A meets B. What line has your crease
made? (ANSWER: a perpendicular bisector) Use a ruler and protractor
to check your answer.
3. Paper folding and circles
Give each student a circle (you can use BLM Circles) and ask students
to draw and label a triangle on their circle.
a) Fold the circle in half so that A meets B.
Geometry 7-5
E-15
Look at the line that the crease in your fold makes. Is it a bisector of
angle C? Is it a perpendicular bisector of line segment AB?
b) Fold the circle in half again, this time making A meet C. What two
properties will the crease fold have?
c) Repeat, making B meet C.
At what point in the circle will all three perpendicular bisectors meet?
Extensions
1. Given a triangle ABC, how can you use perpendicular bisectors to help
you draw the circle going through the points A, B, and C? Draw an acute
scalene triangle. Draw the perpendicular bisectors by hand—do not cut
and fold the triangle.
2. Start with a paper circle. Choose a point C on the circle and draw a right
angle so that its arms intersect the circle. Label the points where the
arms intersect the circle A and B and draw the line segment AB. Repeat
with several circles to produce different right triangles. (You can use
BLM Circles for this Extension.)
A
Fold the circle in two across the side AC so that A falls on C (creating a
perpendicular bisector of AC). Mark the point where B falls on the circle.
Repeat with the side BC, marking the point where A falls on the circle.
What do you notice? (The image of A is the same as the image of B.)
What type of special quadrilateral have you created? (a rectangle)
Repeat the exercise starting with an obtuse or an acute angle C. Do the
images of A and B coincide?
B
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
C
E-16
Teacher’s Guide for Workbook 7.1
G7-6 Parallel Lines
Pages 109–111
Goals
Curriculum
Expectations
Students will identify and draw parallel lines.
Ontario: 7m1, 7m2, 7m46
WNCP: 5SS5, 7SS3,
[C, R, PS]
PRIOR KNOWLEDGE REQUIRED
Can identify and construct a perpendicular using a set square
or a protractor
Can identify and mark right angles
Can name a line segment and identify a named line segment
Can draw and measure with a ruler
Vocabulary
line segment
parallel
right angle
perpendicular
Materials
BLM Distance Between Parallel Lines (p E-46)
BLM Drawing Parallel Lines (p E-44)
Introduce parallel lines—straight lines that never intersect, no matter how
much they are extended. Show how to mark parallel lines with the same
number of arrows.
Have students identify parallel lines in several diagrams. Then have students
identify and mark parallel sides of polygons. Include polygons that have
pairs of parallel sides that are neither horizontal nor vertical.
Introduce the symbol || for parallel lines, label the vertices of the polygons
used above, and have students state which sides are parallel using the new
notation (EXAMPLE: AB || CD).
Ask students to think about where they see parallel lines. Some examples of
parallel lines in the real world are a double centerline on a highway and the
edges of construction beams.
p
?
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
m
process assessment
Workbook Question 8
[R, C], 7m2
n
Determining if two lines are parallel. Have students draw a triangle on grid
paper. Then ask them to draw line segments that are parallel to the sides
of the triangle. For each pair of parallel lines (say, m and n) ask students
to draw a perpendicular (say, p) to one of the lines (m) so that it intersects
both lines. Ask students to predict what the angle between n and p is. Ask
students to explain their prediction. Then have them measure the angle
between the lines. Was the prediction correct?
Have students check the prediction using the other two pairs of parallel lines
they drew. Explain that this property—If one line in a pair of parallel lines
interests a third line at a right angle, the other parallel line also makes a right
angle with the same line—allows us to check whether two lines are parallel
and to construct parallel lines.
Drawing a line segment parallel to a given line segment. Have students
problem-solve how to construct parallel lines using what they’ve just learned
about a perpendicular to parallel lines. As a prompt, you could use the
Geometry 7-6
E-17
process assessment
[PS], 7m1
rectangle and square problems from G7-4 (Extra practice and Bonus,
p E-12—students are required to draw a rectangle or a square using a point
and a pair of parallel lines). ASK: What do you know about the sides of a
rectangle? How does constructing a rectangle mean that you constructed
a pair of parallel lines?
Model drawing the line parallel to a given line through a point using a
protractor (see p 111 in the Workbook or BLM Drawing Parallel Lines), and
then model doing the same thing using a set square. Have students practise
drawing parallel lines using both tools.
ACTIVITIES 1–2
1. Paper folding. Draw a line dark enough so you can see it through
the page. Fold the paper so that you can find a line perpendicular to
AB that does not bisect AB. How can you use this crease to find a line
parallel to AB? How can you use a ruler or any right angle to find a line
parallel to AB?
ANSWER: Fold the paper so that the perpendicular to AB falls onto itself.
The crease is perpendicular to the perpendicular, so it is parallel to AB.
2. Students can investigate distances between parallel lines with BLM
Distance Between Parallel Lines.
Extensions
process assessment
7m1, [PS]
2. Draw:
a) a hexagon with three parallel sides
b) an octagon with four parallel sides
c) a heptagon 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:
a)
E-18
b)
c)
Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
1. A plane is a flat surface. It has length and width, but 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.
Can you find a pair of lines not in a plane that never meet and do
not intersect?
d)
e) f)
3. Lines are parallel if they point in the same direction—that’s why we use
arrows to show parallel lines! We can regard direction as an angle with a
horizontal line. For example, if two lines are both vertical, they both make
a right angle with a horizontal line, and they are parallel. The choice of
a horizontal line as a benchmark is arbitrary—it is just a convention; any
line could be used for that purpose. Indeed, any two lines perpendicular
to a third line are parallel.
Students can complete the Investigation on BLM Properties of Parallel
Lines (p E-45) to learn what happens when parallel lines meet a third
line at different angles.
4. Ask students if they can draw a parallelogram that’s not a rectangle
using only a ruler and a set square. SOLUTION:
Step 2: Draw two of the
parallel sides using the triangle.
Step 3: Use the
ruler to complete
the figure.
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Step 1: Draw
one side of the
parallelogram.
Geometry 7-6
E-19
G7-7 Angle Relationships
Pages 112–113
Goals
Curriculum
Expectations
Ontario: 7m3, 7m7, 7m46,
7m47
WNCP: 6SS2, 7SS1,
[C, R, V]
Students will find angles in triangles, discover the sum of the angles
in a triangle, and use this sum to solve problems.
PRIOR KNOWLEDGE REQUIRED
Can use a protractor to measure angles
Materials
BLM Sum of the Angles in a Triangle (pp E-47–E-48)
Vocabulary
straight angle
acute, obtuse, right angle
adjacent angles
intersecting lines
linear pair
A straight angle is formed when the arms of the angle point in exactly
opposite directions and form a straight line through the vertex of the angle.
Adjacent angles share an arm
and a vertex.
a
c
a
b
f
d
a
b
Two adjacent angles are a
linear pair if, together, they form
a straight angle.
Have students name all the pairs of adjacent angles in the picture at left.
b
ASK: Which angles make a linear pair? Which two combinations of angles
make a straight angle?
1 cup
3
=
1 cup
2
5 cup
6
Point out that the same procedure applies to capacities, volumes, and areas.
4 cm3 + 2 cm3 = 6 cm3
45°
30°
E-20
What happens with angles? Their measures are also added: the measure
of the large angle at left is 30° + 45° = 75°.
Sum of the angles around a point. Ask students to draw a pair of
intersecting lines, measure the angles, and write the measures on the
picture. Then ask them to add up the measures. What is the sum of the
angles around the point? Did they all get the same answer? Show the
Teacher’s Guide for Workbook 7.1
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+
Angle measures in adjacent angles add to the measure of the large
angle. Draw a line segment divided into two smaller segments. Mark the
lengths of the smaller line segments. ASK: What is the length of the whole
line segment? How do you know? What do you do with the lengths of the
smaller line segments to obtain the length of the whole line segment?
53°
127°
53°
128°
picture at left (mention that these angles were made by two intersecting
lines) and ask students how they can tell that whoever measured the angles
made a mistake. (Any two adjacent angles in this picture make a linear pair,
so their sums should all be 180°).
Sum of the angles in a triangle. Have students complete the Investigation
on BLM Sum of the Angles in a Triangle. What is the sum of the angles in
a triangle? Then have students draw several triangles, measure their angles,
and add the measures. Did they get the same sum every time? Explain
that the sums might not add to 180° due to mistakes in measurement. The
protractor, though a convenient tool, is imprecise.
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? (180° minus the sum of the other two angles) Have students find the
measures of the angles in several problems of this sort, then proceed to
more complicated questions, such as the following:
process Expectation
Reflecting on the
reasonableness of
the answer
• All the angles in a triangle are equal. What is the size of each angle?
•O
ne angle of a triangle is 30°. The other two angles are equal. What is
the size of these angles?
• A triangle has two equal angles. One of the angles in this triangle is
90°. What are the sizes of the other two angles?
With the last question, ask whether the equal angles can be 90° each.
Have students explain why this is not possible. (The two angles would
already add to 180°, leaving no room for the third angle.) Invite a volunteer to
draw the correct triangle on the board and mark the measures of the angles.
Then present a similar problem:
• A triangle has two equal angles. One of the angles in this triangle
is 50°. What are the sizes of the other two angles?
ASK: How is this problem different from the previous problem? (The given
process Expectation
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Reflecting on other ways
to solve the problem
process assessment
Workbook, Question 11
[C, V], 7m7
angle is an acute angle, not a right angle.) Can a triangle have two angles of
50°? What is the third angle then? (80°) Draw an acute isosceles triangle on
the board and ask volunteers to mark the angles on the picture. Then draw
another acute isosceles triangle, mark the base angles as equal, and mark
the unequal angle as 50°. Can this situation happen? What are the measures
of the other two angles?
Extra practice:
a) If half an angle is 20°, the whole b) If one-third of an angle is 30°, angle is ______°.
the whole angle is ______°.
20°
Geometry 7-7
30°
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c) What is half of 90°? ______°
d) If one-quarter of an angle is 25°, the whole angle is ______°.
25°
?
Bonus
What is one-third of 120°? ______°
ACTIVITIES 1–2
1. Create right angles by paper folding
Give your students a piece of paper that has no corners (e.g., an oval
or a cloud shape). Ask students to create a right angle on the paper by
folding. (POSSIBLE ANSWER: Fold once, unfold to see your straightline crease, then fold again across the crease so that the sides of the
crease coincide.)
ASK: When you folded the paper the first time, what is the size of the
process assessment
7m7, [C]
angle you created? (180°, a straight angle) What fraction of the straight
angle is the right angle? (one half) Use your answers to these two
questions to explain why your angle is a right angle. (ANSWER: A right
angle is 90°, so two right angles are 180°, or a straight angle. A right
angle is exactly half of a straight angle. In other words, when you fold
the paper the second time and the halves of the first crease coincide,
you know that the two angles you have created are equal. Together,
the angles make a straight angle, so they have to be right angles.)
2. Start with a cloud- or oval-shaped piece of paper. How can you fold
the paper to create the following?
Students can work independently or in pairs to try and create the angle
and shapes listed. Invite students to share strategies and solutions
with the class, then, if necessary, give students the correct sequence of
steps to follow. Do not unfold the paper unless it is mentioned as part
of instructions. The instructions might be easier to understand if you
actually perform the folding as you go along.
ANSWERS:
a) Make a 45° angle:
Create a right angle by the method of Activity 1. Fold the right angle
through the vertex so that its arms fall onto each other. This divides the
angle into two equal parts, so the angle is a 45° angle.
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Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
a) a 45° angle
b) a square
c) a rectangle
b) Make a square:
i) Fold the paper to make
a line segment. Fold the line
segment onto itself to create
a right angle; the vertex of
the right angle will be one of
the vertices of the square.
ii) Fold the right angle in
two to make a 45° angle.
Sides of square
iii) Fold the corner of the
45° angle onto one of the
arms, so that parts of that
arm meet each other. Trace
the fold you created with
a pencil.
iv) Unfold once and trace
the crease that is revealed
with a pencil.
v) Unfold once again and flip
the paper over. The lines you
drew form two more sides of
the square.
c) Make a rectangle:
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
Geometry 7-7
i) Repeat step i) from part b).
Sides of rectangle
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ii) Choose a point on one of
the arms of your right angle
to be the second vertex of
the rectangle. Fold the paper
through the chosen point so
that the arm of the right angle
you created folds onto itself.
This will be the third side of the
rectangle.
Second corner
of the rectangle
Third side of
the rectangle
COPYRIGHT © 2010 JUMP MATH: NOT TO BE COPIED
iii) Flip the paper over. Fold
the bottom part of the strip
backwards so that both creases
forming the sides of the strip
fold onto themselves.
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Teacher’s Guide for Workbook 7.1
NAME
DATE
Measuring and Drawing Angles and Triangles
Measuring an angle
30°
arm
origin
If the arms are
too short to reach
the protractor scale,
lengthen them.
Step 1: Place the
origin of the protractor
over the vertex of
the angle.
Step 2: Rotate the
protractor so the base
line is exactly along
one of the arms of
the angle.
0°
0° 180°
base line
Step 3: Look at that
arm of the angle and
choose the scale that
starts at 0°.
Step 4: Use that
scale to find the
measurement.
Drawing an angle
angle mark
angle mark
60°
Step 1: Draw a line
segment.
Step 2: Place the protractor
with the origin on one endpoint.
This point will be the vertex of
the angle.
Step 3: Hold the protractor in
place and mark a point at the
angle measure you want.
Step 4: Draw a line
from the vertex through
the angle mark.
Drawing lines that intersect at an angle
45°
P
Step 1: Draw a line. Mark a point P
on the line.
45°
P
Step 2: Draw an angle of the given
measure using P as vertex.
P
Step 3: Extend the arms of your angle
to form lines.
90° 30°
5 cm
90°
5 cm
5 cm
Step 1: Sketch the
Step 2: Use a ruler to draw
triangle you want to draw. one side of the triangle.
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90° 30°
5 cm
Step 3: Use a protractor to draw
the angles at each end of this side.
Extend the arms until they intersect.
90° 30°
5 cm
Step 2: Erase any extra
arm lengths.
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Drawing a triangle
NAME
DATE
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Circles
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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NAME
DATE
Distance Between Parallel Lines
InveStIgatIon
Does the distance between parallel lines depend on where you measure it?
a. Measure the line segments with endpoints on the two parallel lines with a ruler.
Write the lengths of the line segments on the picture.
B. Use a square corner to draw at least three perpendiculars from one parallel line to
the other, as shown.
Measure the distance between the two parallel lines along the perpendiculars.
What do you notice?
C. Explain why all the perpendiculars you drew in part B are parallel.
e. Measure the line segments you drew between the two given parallel lines in part D.
What do you notice?
F. To measure the distance between two parallel lines, draw a line segment
perpendicular to both lines and measure it. Does the distance between parallel lines
depend on where you measure it?
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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D. A parallelogram is a 4-sided polygon with opposite sides parallel. You can draw
parallelograms by using anything with parallel sides, like a ruler. Place a ruler across
both of the parallel lines and draw a line segment along each side of the ruler. Use
this method to draw at least 3 parallelograms with different angles.
NAME
DATE
Drawing Parallel Lines
Drawing a line parallel to AB through point P
Using a set square
P
A
P
P
A
B
Step 1: Line up one of the
short sides of the set square
with AB.
P
A
B
A
Step 2: Use the set square
and a straight-edge to draw
a perpendicular to AB.
B
B
Step 3: Draw a line
perpendicular to the new
line that passes through P.
Step 4: Erase the line you
no longer need.
Using a protractor
P
A
P
P
B
A
B
B
A
B
Step 2: Line up the 90° line on the protractor with the
line segment drawn in step 1, and the straight side of the
protractor with point P. Draw a line parallel to AB. Erase
the first perpendicular you drew.
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Step 1: Line up the 90° line on the protractor with AB.
Use the straight side of the protractor to draw a line
segment perpendicular to AB.
A
P
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Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
NAME
DATE
Properties of Parallel Lines
Investigation What happens if two lines meet a third line at the
same angle, but it is not a right angle?
A. Draw a pair of parallel lines and a third line intersecting both at
an angle that is not a right angle.
B. ∠ABD = ∠ACE = 70°. Draw a perpendicular to BD through point A.
Extend it to meet CE. Is the line you drew perpendicular to CE? Check
using a protractor. What can you say about the lines BD and CE?
E
D
C
B
A
D
C. ∠ABD = ∠ACE = 70°. Are the lines BD and CE parallel?
E
B
A
C
D. Draw a pair of lines that intersect at a 40° angle. Draw a third line
that meets one of the lines at the same angle. Try to make the
third line parallel to one of the lines you started with. Check by
drawing a perpendicular.
E. Compare the pattern between the equal angles ∠ABD and ∠ACE
in parts B and D. Which one looks more like the angles marked
in the letter C and which one is more like angles in the letter F?
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Mathematicians have proved that if two lines meet with a third line
at the same angles creating a pattern like in the letter F, the lines
are parallel.
When the lines meet at a right angle, you do not have to worry about
the pattern of equal angles—they are all right angles.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
C
E
B
D
A
C
E
A
B
D
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NAME
DATE
Sum of the Angles in a Triangle
Investigation
What is the sum of the angles in a triangle?
A. Circle the combinations of a 70° angle and another angle that will make a triangle.
(Hint: Imagine the sides of the triangle extended—will they ever intersect?)
70°
80°
70°
90°
70°
100°
70°
110°
70°
120°
Circle the combinations of a 50° angle and another angle that will make a triangle.
50°
100°
50°
110°
50°
120°
50°
130°
50°
140°
Circle the combinations of a 90° angle and another angle that will make a triangle.
90°
70°
90°
80°
90°
90°
90°
100°
90°
Make a prediction:
To make a triangle, the total measures of any two angles must be less than °.
110°
B. List the sum of the measures of the angles in each triangle.
90°
° + ° + ° = °
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118°
55°
68°
42°
57°
20°
70°
° + ° + ° = °
25°
20°
25°
130°
° + ° + ° = ° ° + ° + ° = °
What do you notice about the sums of the angles? Do you think this result will be true for all triangles? Make a conjecture: The sum of the three angles in any triangle will always be °.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
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NAME
DATE
C. Calculate the sum of the angles.
70°
°+
90°
°+
42°
57°
20°
°=
°
°+
68°
55°
°+
°=
25°
25°
118°
°
130°
20°
°+
°+
°=
°
°+
°+
°=
°
What do you notice about the sums of the angles?
D. Cut out a paper triangle and fold it as follows:
C
A
B
Step 1: Find the midpoints of the
sides adjacent to the largest angle
(measure or fold). Draw a line
between the midpoints.
A
B
C
Step 2: Fold the triangle along the
new line so that the top vertex meets
the base of the triangle. You will get
a trapezoid.
A
C
B
Step 3: Fold the other two vertices
of the triangle so that they meet the
top vertex.
The three vertices folded together add up to a straight angle.
What is the sum of the angles in a straight angle?
°
e. Could you fold the vertices of any triangle along a line and get a straight angle as
you did in part D? Do the results of the paper folding activity support your conjecture
in part B? Explain.
F. In fact, it has been mathematically proven that…
the sum of the angles in a triangle is
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°.
Blackline Master — Geometry — Teacher’s Guide for Workbook 7.1
COPYRIGHT © 2010 JUMP MATH: TO BE COPIED
So ∠A + ∠B + ∠C =
°