6.4-6.5: Similarity Shortcuts
Objectives:
1. To discover and use
shortcuts for
determining that two
triangles are similar
2. To find missing
measures in similar
polygons
Assignment:
• P. 384-387: 1-4, 7, 8,
10, 12, 14-17, 20, 30,
31, 32, 36, 41, 42
• P. 391-395: 4, 6-8,
10-14, 33, 39, 40
• Challenge Problems
OBJECTIVE 1
You will be able to
discover and use
shortcuts for determining
that two triangles are
similar
Warm-Up
Since they are polygons, what two things
must be true about triangles if they are
similar?
Similar Polygons
Two polygons are similar polygons iff the corresponding
angles are congruent and the corresponding sides are
proportional.
Similarity Statement:
N
C
𝐶𝑂𝑅𝑁~𝑀𝐴𝐼𝑍
Corresponding Angles:
N
C
∠𝐶 ≅ ∠𝑀 ∠𝑂 ≅ ∠𝐴
∠𝑅 ≅ ∠𝐼 ∠𝑁 ≅ ∠𝑍
O
M
Z
R
A
O
Statement
of Proportionality:
R
𝐶𝑂 𝑂𝑅 𝑅𝑁 𝑁𝐶
=
=
=
𝑀𝐴 𝐴𝐼
𝐼𝑍
𝑍𝑀
A
I
Example 1
Triangles ABC
and ADE are
similar. Find the
value of x.
D
B
A
6 cm
9 cm
8 cm
C
x
E
Example 2
Are the triangles below similar?
8
4
6
3
37
53
5
10
Do you really have to check all the sides and angles?
Investigation 1
In this Investigation we will check the first
similarity shortcut. If the angles in two
triangles are congruent, are the triangles
necessarily similar?
F
C
A
50
40
B
D
50
40
E
Investigation 1
Step 1: Draw Δ𝐴𝐵𝐶 where 𝑚∠𝐴 and 𝑚∠𝐵
equal sensible values of your choosing.
C
A
50
40
B
Investigation 1
Step 1: Draw Δ𝐴𝐵𝐶 where 𝑚∠𝐴 and 𝑚∠𝐵
equal sensible values of your choosing.
Step 2: Draw Δ𝐷𝐸𝐹 where 𝑚∠𝐷 = 𝑚∠𝐴 and
𝑚∠𝐸 = 𝑚∠𝐵 and 𝐴𝐵 ≠ 𝐷𝐸.
F
C
A
50
40
B
D
50
40
E
Investigation 1
Now, are your triangles similar? What would
you have to check to determine if they are
similar?
F
C
A
50
40
B
D
50
40
E
Angle-Angle Similarity Postulate
If two angles of one
triangle are
congruent to two
angles of another
triangle, then the two
triangles are similar.
Example 3
Determine whether the triangles are similar.
Write a similarity statement for each set of
similar figures.
Investigation 3
Each group will be given one
of the three candidates for
similarity shortcuts. Each
group member should start
with a different triangle and
complete the steps outlined
for the investigation.
Share your results and
make a conjecture based
on your findings.
Side-Side-Side Similarity Theorem
If the corresponding side lengths of two triangles
are proportional, then the two triangles are
similar.
Side-Angle-Side Similarity Theorem
If two sides of one triangle are proportional to
two sides of another triangle and the included
angles are congruent, then the two triangles are
similar.
Example 4
Are the triangles below similar? Why or why
not?
Objective 2
You will be able to find
missing measures in
similar polygons
Indirect Measurement
Indirect
measurement
involves
measuring
distances that
cannot be easily
measured directly.
This often involves
using properties of
similar triangles.
Thales
The Greek mathematician
Thales was the first to
measure the height of a
pyramid by using
geometry. He showed
that the ratio of a
pyramid to a staff was
equal to the ratio of one
shadow to another.
Example 5
If the shadow of the pyramid is 576 feet, the
shadow of the staff is 6 feet, and the height
of the staff is 5 feet, find the height of the
pyramid.
Example 6
Explain why Thales’ method worked to find
the height of the pyramid?
Example 7
If a person 5 feet tall casts a 6-foot shadow
at the same time that a lamppost casts an
18-foot shadow, what is the height of the
lamppost?
Investigation 3
What if you decide to
indirectly measure a
height on a day when
there are no shadows?
The following GSP
Animation will help you
discover an alternate
method of indirect
measurement using a
mirror.
Example 8
Your eye is 168
centimeters from
the ground and you
are 114 centimeters
from the mirror.
The mirror is 570
centimeters from
the flagpole. How
tall is the flagpole?
Example 9
Find the values of 𝑥 and 𝑦.
28
24
24
x
18
y
6.4-6.5: Similarity Shortcuts
Objectives:
1. To discover and use
shortcuts for
determining that two
triangles are similar
2. To find missing
measures in similar
polygons
Assignment:
• P. 384-387: 1-4, 7, 8,
10, 12, 14-17, 20, 30,
31, 32, 36, 41, 42
• P. 391-395: 4, 6-8,
10-14, 33, 39, 40
• Challenge Problems