Download Protractors

NAME
DATE
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Protractors
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-89
DATE
NAME
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.
base line
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°
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.
O-90
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.
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Drawing a triangle
DATE
NAME
Drawing Perpendicular Lines and Bisectors
Drawing a line segment perpendicular to AB through point P
Using a set square
P
A
P
B
A
P
B
P
A
A
B
Here point P is on AB.
B
Here point P is outside AB.
Using a protractor
P
A
B
P
A
P
B
P
B
A
Here point P is on AB.
A
B
Here point P is outside AB.
Drawing the perpendicular bisector of line segment AB
Using a set square or protractor
M
A
A
B
A
Step 1: Use a ruler to
determine the midpoint of
the line segment. Label it M.
M
B
A
B
M
B
M
Step 2: Use a set square or a protractor to draw a
line perpendicular to AB that passes through M.
The line you have drawn
is the perpendicular bisector
of AB.
Using a compass and a straightedge
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
ℓ
A
Step 1: Put your compass point
at one end of the line segment.
Construct an arc as shown.
B
A
B
Step 2: With your compass at the same
radius, construct a second arc centred
at B as shown.
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
A
B
Step 3: Construct a line ℓ through the
intersection points of the two arcs. This
is the perpendicular bisector of AB.
O-91
DATE
NAME
Drawing Parallel Lines
Drawing a line parallel to AB through point P
Using a set square
P
A
P
A
B
B
A
Step 2: Use the set square
and a straightedge to draw
a perpendicular to AB.
Step 1: Line up one of the
short sides of the set square
with AB.
P
P
A
B
Step 3: Draw a line
perpendicular to the new
line that passes through P.
B
Step 4: Erase the line you
no longer need.
Using a protractor
P
A
P
P
B
A
B
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
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.
Using a compass and a straightedge
ℓ
ℓ
m
A
B
A
Step 1: Mark any two points A, B on the line. Construct the
perpendicular bisector of AB.
O-92
D
B
Step 2: Mark any two points C, D on the line you drew.
Construct the perpendicular bisector of CD. Label it m.
Line m is parallel to AB.
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
C
NAME
DATE
Distance Between Parallel Lines
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.
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.
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
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 draw the perpendicular? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-93
DATE
NAME
Sum of the Angles in a Triangle (1)
What is the sum of the angles in a triangle?
INVESTIGATION
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°
50°
90°
70°
100°
70°
110°
70°
120°
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°
Circle the combinations of a 50° angle and another angle that will make a triangle.
80°
70°
90°
80°
90°
90°
90°
100°
ake a prediction: M
To make a triangle, the total measures of any two angles must be less than
90°
110°
°.
B. List the sum of the measures of the angles in each triangle.
57°
70°
90°
69°
° +
° +
° =
°
° +
° +
°
° =
°
24°
41°
116°
° +
23°
24°
° +
° =
°
° +
132°
° +
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 °.
O-94
° =
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
54°
20°
DATE
NAME
Sum of the Angles in a Triangle (2)
C. Calculate the sum of the angles.
70°
90°
20°
57°
° + ° + ° = °
41°
54°
° + ° + ° = °
116°
24°
24°
23°
69°
132°
° + ° + ° = °
° + ° + ° = °
What do you notice about the sums of the angles? D. Cut out a paper triangle and fold it as follows:
C
A
Step 1: Find the midpoints of the
sides adjacent to the largest angle
(measure or fold). Draw a line
between the midpoints.
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
B
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.
The three vertices folded together add up to a straight angle.
What is the sum of the angles in a straight angle?
So ∠A + ∠B + ∠C =
A
C
B
Step 3: Fold the other two vertices
of the triangle so that they meet the
top vertex.
°
°
E.Could you fold the vertices of any triangle as you did in part D and get a straight
angle? Do the results of the paper folding 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
°.
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-95
NAME
DATE
Quadrilaterals (1)
2
3
4
5
6
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
1
O-96
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
NAME
DATE
Quadrilaterals (2)
8
9
10
11
12
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
7
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-97
DATE
NAME
Sorting Quadrilaterals
Properties of sides
Shapes with
the property
Properties of angles
No equal sides
No equal angles
1 pair of equal sides
1 pair of equal angles
2 pairs of equal sides
2 pairs of equal angles
Equal sides are adjacent
Equal angles are adjacent
Equal sides are opposite
Equal angles are opposite
4 equal sides
4 equal angles
1 pair of parallel sides
No pairs of angles add to 180°
2 pairs of parallel sides
2 pairs of angles add to 180°
Perpendicular diagonals
4 pairs of angles add to 180°
Equal diagonals
6 pairs of angles add to 180°
Properties of sides
Shapes with
the property
Properties of angles
Equal diagonals
One of the diagonals is also
an angle bisector
Exactly one diagonal bisects
the other
Both diagonals are
angle bisectors
Both diagonals bisect each other
Shapes with
the property
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Perpendicular diagonals
Shapes with
the property
O-98
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
NAME
DATE
Triangles
Set 1
Set 2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Set 3
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-99
DATE
NAME
Triangles for Sorting
B
A
E
D
C
I
G
H
F
L
J
M
O-100
N
O
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
K
DATE
NAME
Two Pentagons
O
N
A
Q
R
M
B
K
L
P
S
T
a) How many right angles does each pentagon have? b)Cut the pentagons out. Fold the pentagons so that you can see that the remaining
angles are all equal.
c) Does each side on pentagon A have a side of the same length on pentagon B? d)Match each side from pentagon A with a side of the same length from pentagon B.
Do not use the same side twice.
KL = LM = MN = NO = OK = COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
e) If you place one pentagon on top of the other, do they match? f) Are they of the same shape? g) Can we say that these pentagons have the same sides and angles? h) Are the pentagons congruent? i) Explain why these pentagons have the same sides and angles but aren’t congruent.
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-101
DATE
NAME
Investigating Congruence
INVESTIGATION Do you have to check that all 3 pairs of sides and all 3 pairs of
angles are equal to determine if two triangles are congruent?
3 sides
Conjecture: If two triangles have 3 pairs of equal sides, then the triangles
are congruent.
Test the conjecture: Take 3 straws of different lengths. Make a triangle with
them and trace it. See if you can make a different triangle with the same
three straws. Try with a different set of three straws.
Are your triangles congruent? Do you think this result will be true for all triangles? 2 sides and the angle between
Conjecture: If 2 sides and the angle between them in one triangle are equal to 2 sides
and the angle between them in another triangle, the triangles are congruent.
Test the conjecture: Draw 3 triangles, each with one side 3 cm long, one side
5 cm long, and a 45° angle between these sides.
Are the triangles congruent? Do you think this result will be true for all triangles?
Pick three measurements: a = cm, b = cm, C = °
Draw three triangles, each with one side a cm long, one side b cm long, and a C° angle
between these sides. Are your triangles congruent?
2 angles and the side between them
Test this conjecture: Draw 3 triangles, each with a 60° angle, a 45° angle,
and a side 5 cm long between these angles.
Are the triangles congruent? Do you think this result will be true for all triangles? Pick three measurements: a = cm, B = °, C = °
Draw three triangles, each with a B° angle, a C° angle, and a side a cm long between
these angles. Are your triangles congruent? O-102
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Conjecture: If 2 angles and the side between them in one triangle are equal to 2 angles
and the side between them in another triangle, the triangles are congruent.
DATE
NAME
Congruence Rules on Geometer’s Sketchpad (1)
1.Using the polygon tool, construct a triangle, ABC. Measure the sides and the angles
of your triangle.
2. a) Construct a point D. Using a command for constructing circles and the length
of AB as the radius, construct a circle with centre D. Construct line segment
DE = AB. Hide the circle.
b) Using a command for constructing circles and the length of BC as the radius,
construct a circle with centre E.
c) Using a command for constructing circles and the length of AC as the radius,
construct a circle with centre D.
d) Construct a point that is on both circles. Label it F. Use the polygon tool to
construct triangle DEF. Hide the circles.
3. a) Which sides are equal in ∆ABC and ∆DEF?
= , = , = b) Measure the angles of ∆DEF. What can you say about ∆ABC and ∆DEF?
4. Try to move the vertices of ∆DEF around.
a) How does your triangle change? Which transformations can you make: rotations,
reflections, translations?
b) Can you move ∆DEF onto ∆ABC to check whether they are congruent? Do you need to reflect the triangle to do that? COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around.
a) Are the changes you can make in this triangle different from the changes you
could make to ∆DEF? Why?
b) What happens to ∆DEF when you modify ∆ABC? What can you say about the
triangles ∆ABC and ∆DEF?
6. Are three side lengths enough to determine a unique triangle? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-103
DATE
NAME
Congruence Rules on Geometer’s Sketchpad (2)
1.Using the polygon tool, construct a triangle, ABC. Measure the sides and the angles
of your triangle.
2. a) Construct a point D. Using a command for constructing circles and the length
of AB as the radius, construct a circle with centre D. Construct line segment
DE = AB. Hide the circle.
b) Using a command for constructing circles and the length of BC as the radius,
construct a circle with centre E.
c) Select E as a centre of rotation. Use a command in the Transformation menu to
mark ∠ABC as the angle of rotation. Rotate point D around E by the angle
chosen. Construct a ray from E through the image of D.
d) Construct a point that is on the ray and the circle. Label it F. Use the polygon tool
to construct triangle DEF. Hide the circle and the ray EF.
3. a) Which sides and angles are equal in ∆ABC and ∆DEF?
= , = , = b) Measure the rest of the sides and angles of ∆DEF. What can you say about
∆ABC and ∆DEF?
4. Try to move the vertices of ∆DEF around.
a) How does your triangle change? Which transformations can you make: rotations,
reflections, translations?
b) Can you move ∆DEF onto ∆ABC to check whether they are congruent? Do you need to reflect the triangle to do that? a) Are the changes you can make in this triangle different from the changes you
could make to ∆DEF? Why?
b) What happens to ∆DEF when you modify ∆ABC? What can you say about the
triangles ∆ABC and ∆DEF?
6. Are two sides and the angle between them enough to determine a unique triangle? O-104
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around.
DATE
NAME
Congruence Rules on Geometer’s Sketchpad (3)
1. Using the polygon tool, construct a triangle, ABC. Measure the sides and the angles
of your triangle.
2. a) Construct a point D. Using a command for constructing circles and the length
of AB as the radius, construct a circle with centre D. Construct line segment
DE = AB. Hide the circle.
b) Select E as a centre of rotation. Use a command in the Transformation menu
to mark ∠ABC as the angle of rotation. Rotate point D around E by the angle
chosen. Construct a ray from E through the image of D.
c) Select D as a centre of rotation. Mark ∠BAC as the angle of rotation. Rotate point
E around D by the angle chosen. Construct a ray from D through the image of E.
d) Construct a point F that is the intersection point of both rays. Use the polygon
tool to construct triangle DEF. Hide the rays.
3. a) Which sides and angles are equal in ∆ABC and ∆DEF?
= , = , = b) Measure the rest of the sides and angles of ∆DEF. What can you say about
∆ABC and ∆DEF?
4. Try to move the vertices of ∆DEF around.
a) How does your triangle change? Which transformations can you make: rotations,
reflections, translations?
b) Can you move ∆DEF onto ∆ABC to check whether they are congruent? Do you need to reflect the triangle to do that? COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
5. Move ∆DEF away from ∆ABC. Try to move the vertices of ∆ABC around.
a) Are the changes you can make in this triangle different from the changes you
could make to ∆DEF? Why?
b) What happens to ∆DEF when you modify ∆ABC? What can you say about the
triangles ∆ABC and ∆DEF?
6. Are two angles and a side between them enough to determine a unique triangle? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-105
DATE
NAME
Proving the Pythagorean Theorem (1)
REMINDER
Area of triangle = base × height ÷ 2
1. PQRS is a rectangle.
a) Find the areas of the shapes:
i) PQRS
R 3 cm
Q
ii) ∆PRS
T
5 cm
iii) ∆PTS
P
b) Describe how the three areas in a) are related.
7 cm
S
2. CDEA, AFGB, and CBHJ are squares.
a) The base of the two shapes is given. Find the heights of the shapes.
i) ∆CBD
ii) ∆CBG CDEA
base: DC base: DC
height: AC height: AC AFGB
base: BG
base: BG
height: height: E
D
F
A
A
G
iii) ∆CAJ
B
iv) ∆HBA LKJC
base: JC
base: JC
height: height: BHKL
base: BH
base: BH
height: height: A
C
J
A
K
C
B
L
H
J
L
K
B
H
b) Write the area of each triangle in terms of another shape.
1
1
i) Area of ∆CBD =
of area of CDEA ii) Area of ∆CBG =
of area of 2
2
O-106
iii) Area of ∆CAJ =
1
of area of 2
iv) Area of ∆HBA =
1
of area of 2
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
C
B C
DATE
NAME
Proving the Pythagorean Theorem (2)
3.Fill in the blanks to finish proving the Pythagorean Theorem the same way Euclid
did it 2300 years ago.
In a triangle ABC with ∠A = 90°, AB2 + AC2 = BC2.
a) The picture at right shows the Pythagorean Theorem.
What special quadrilateral is each of these quadrilaterals?
CDEA square AFGB CBHJ LKJC BHKL b) Area of ∆CBD = of area of square CDEA, because
e) ∆CBD ≅ ∆CAJ by congruence rule, because
DC = , CB = ∠DCB = 90° + ∠ = ∠ J
K
H
E
g)Area of ∆CBG = of area of square , because
they both have base and height .
h)Area of ∆HBA = of area of rectangle , because
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
B
L
f) Area of CDEA = area of LKJC because D
F
A
they both have base and height .
i)Mark the equal line segments in the picture.
j) ∆CBG ≅ by congruence rule, because
BG = , ∠CBG = 90° + ∠ = ∠ G
C
they both have base CJ and height .
d) Mark the equal line segments in the picture.
F
A
they both have base and height .
D
c)Area of ∆CAJ = of area of rectangle , because
E
G
C
L
B
CB = J
H
K
k) Area of AFGB = area of BHKL because j) AB2 + AC2 = BC2, because Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-107
NAME
DATE
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Circles
O-108
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
DATE
NAME
Angle Properties
Supplementary angles are angles that add to 180°.
1
2
A
60°
B
3
120°
4
Corresponding angles make a pattern similar to the letter F.
l
l
1
l
2
m
n
5
m
m
3
n
6
l
m
4
n
n
8
7
Corresponding Angle Theorem:
When lines are parallel, corresponding angles are equal.
The reverse statement is true as well:
When corresponding angles are equal, the lines are parallel.
Alternate angles make a pattern similar to the letter Z.
l
l
3
m
6
4
p
5
n
r
Alternate Angle Theorem:
When lines are parallel, alternate angles are equal.
The reverse statement is true as well:
When alternate angles are equal, the lines are parallel.
Opposite angles are created when two lines intersect.
1
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Opposite Angle Theorem:
Opposite angles are equal: ∠1 = ∠2 and ∠3 = ∠4
4
3
2
Complementary angles are angles whose sum is 90°.
B
A
x
60°
30°
y
1
2
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-109
DATE
NAME
Properties of Angles in a Triangle
Sum of Angles in a Triangle Theorem
The sum of the angles in a triangle is 180°.
∠A + ∠B + ∠C = 180°
B
A
C
Exterior Angle Theorem
The measure of an exterior angle in a triangle is equal to the sum of the opposite angles
in the triangle.
∠x = ∠a + ∠b
b
a
c
x
Isosceles Triangle Theorem
The base angles in an isosceles triangle are equal.
The reverse statement is true as well:
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
If two angles in a triangle are equal, then the triangle is isosceles, with the equal sides
adjacent to the third angle in the triangle.
O-110
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
NAME
DATE
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Rectangles
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-111
DATE
NAME
Similarity Rules Using Geometer’s Sketchpad (1)
INVESTIGATION
are similar?
Which information about a pair of triangles will ensure they
A. SSS (side-side-side)
a) Draw a triangle, ABC. Measure the sides.
b) Create a new parameter, s, to be the scale factor. Set s = 2.
c) Construct a triangle A′B′C′ with sides A′B′ = sAB, A′C′ = sAC, B′C′ = sBC.
d) Measure the angles of both triangles. What do you notice?
Are the triangles similar? e) Modify ∆ABC by moving the vertices.
Does ∆A′B′C′ stay similar to ∆ABC? f) Conjecture: SSS is a similarity rule:
A ' B ' A 'C ' B 'C '
If
=
=
, then ∆ABC is similar to ∆A′B′C′.
AB
AC
BC
Change the scale factor (s) to three different values, including two decimals, one
of them more than 1 and the other less than 1.
Does ∆A′B′C′ stay similar to ∆ABC? Is the conjecture true? B. SAS (side-angle-side )
a) Draw a triangle, ABC. Measure the sides and the angles.
b) Create a new parameter, s, to be the scale factor. Set s = 3.
c) Construct a triangle A′B′C′ with sides A′B′ = sAB, A′C′ = sAC and ∠A′ = ∠A.
d)Measure the side B′C′ and the angles ∠B′, ∠C′. Find the ratio B′C′ : BC. What do
you notice?
Are the triangles similar? e)Modify ∆ABC by moving the vertices. Does the triangle ∆A′B′C′ stay similar
to ∆ABC? f) Conjecture: SAS is a similarity rule:
A ' B ' A 'C '
If ∠A = ∠A′ and
=
, then ∆ABC is similar to ∆A′B′C′.
AB
AC
Change the scale factor (s) to three different values, including two decimals,
one of them more than 1 and the other less than 1.
Does ∆A′B′C′ stay similar to ∆ABC? Is the conjecture true? O-112
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
DATE
NAME
Similarity Rules Using Geometer’s Sketchpad (2)
C. ASA (angle-side-angle)
a) Draw a triangle, ABC. Measure the sides and the angles.
b) Create a new parameter, s, to be the scale factor. Set s = 4.
c) Construct a triangle A′B′C′ with sides A′B′ = sAB, ∠A′ = ∠A and ∠B′ = ∠B.
d)Measure the sides B′C′, A′C′ and the angle ∠C′. Find the ratios B′C′ : BC and
A′C′ : AC. What do you notice?
Are the triangles similar? e) Modify ∆ABC by moving the vertices.
Does the ∆A′B′C′ stay similar to ∆ABC? D. The ratios between corresponding sides of similar triangles are the same. In which
parts—A, B or C—did you construct the triangles so that at least two pairs of
corresponding sides had the same ratios? How was the construction in the other part different?
a) Draw a triangle, ABC. Measure the sides and the angles.
b)Construct a triangle A′B′C′ with ∠A′ = ∠A and ∠B′ = ∠B. What do you know
about the angles C and C′? Explain. A ' B ' C ' B ' A 'C '
c)Measure the sides of ∆A′B′C′ . Find the ratios
,
,
. What do
AB CB AC
you notice?
Are the triangles similar? d) Modify ∆ABC by moving the vertices.
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Does ∆A′B′C′ stay similar to ∆ABC? e) Conjecture: AA (angle-angle) is a similarity rule:
If ∠A = ∠A′ and ∠B = ∠AB′, then ∆ABC is similar to ∆A′B′C′.
Modify ∆A′B′C′ by moving the vertices. What do you notice about the ratios
between the corresponding sides?
Does ∆A′B′C′ stay similar to ∆ABC? Is the conjecture true? Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
O-113
DATE
NAME
Area of Parallelogram and Triangle
Parallelogram
height
bases
A parallelogram is a quadrilateral with two pairs of parallel sides.
Any pair of parallel sides can be chosen to be the bases.
The distance between these two parallel sides is the height.
The height is measured along a line perpendicular to the bases. This line can be drawn
anywhere. In these pictures, the thick black line is one of the bases and the dashed line
is the height.
Area of a parallelogram = base × height
Triangle
height
base
A triangle is a polygon with three sides.
Any side of a triangle can be the base. Draw a perpendicular from the vertex opposite
the base to the base. The distance from the vertex to the base along that perpendicular
is the height.
Any triangle is half of a parallelogram with the same base and height.
Area of a triangle = base × height ÷ 2
O-114
Blackline Master — Geometry — Teacher’s Guide for Workbook 8.2
COPYRIGHT © 2011 JUMP MATH: TO BE COPIED
Sometimes the height is outside the triangle. In these pictures, the thick black line is the
base and the dashed line is the height.