18-1: Similarity Criteria
Objectives:
1. To discover and use
shortcuts for
determining that two
triangles are similar
Assignment:
• P. 261: 10-14
• P. 271: 1
• 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?
18-1: Similarity Criteria
Objectives:
1. To discover and use
shortcuts for
determining that two
triangles are similar
Assignment:
• P. 261: 10-14
• P. 271: 1
• Challenge Problems