1. 9.3 Similar Triangles

2. Vocabulary What You'll Learn Similar Triangles
You will learn to use AA, SSS, and SAS similarity tests for
triangles.
Vocabulary
Nothing New!

3. Similar Triangles
Some of the triangles are similar, as shown below.
The Bank of China building in Hong Kong is one of the ten tallest buildings in
the world.
Designed by American architect I.M. Pei, the outside of the 70-story building
is sectioned into triangles which are meant to resemble the trunk of a bamboo
plant.

4. Similar Triangles
In previous lessons, you learned several basic tests for determining whether
two triangles are congruent. Recall that each congruence test involves only
three corresponding parts of each triangle.
Likewise, there are tests for similarity that will not involve all the parts of
each triangle.
Postulate
9-1
AA Similarity
If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are ______.
similar C F A D E B
If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF

5. If the measures of the sides of a triangle are ___________
Similar Triangles
Two other tests are used to determine whether two triangles are similar.
Theorem 9-2
SSS Similarity
If the measures of the sides of a triangle are ___________
to the measures of the corresponding sides of another triangle, then the triangles are similar.
proportional C 6 F 2 3 1 A B D 4 E 8
then the triangles are similar
then ΔABC ~ ΔDEF

6. If the measures of two sides of a triangle are ___________
Similar Triangles
Theorem 9-3
SAS Similarity
If the measures of two sides of a triangle are ___________
to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar.
proportional C F 2 1 A B D 4 E 8
then ΔABC ~ ΔDEF

7. is used and complete the statement.
Similar Triangles
Determine whether the triangles are similar. If so, tell which similarity test
is used and complete the statement. G H K M P J 6 14 10 9 15 21 6 10 14
Since
, the triangles are similar by SSS similarity. = = 9 15 21
Therefore, ΔGHK ~ Δ
JMP

8. If Fransisco is 6 feet tall, how tall is the tree?
Similar Triangles
Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet long at the same time that his shadow is 4 feet long.
If Fransisco is 6 feet tall, how tall is the tree?
1) The sun’s rays form congruent angles with the ground.
2) Both Fransisco and the tree form right angles with the ground. 4 6 = t 18
4t = 108
t = 27
6 ft.
The tree is 27 feet tall!
4 ft.
18 ft.

9. Similar Triangles Slade is a surveyor.
To find the distance across Muddy Pond, he forms similar triangles and measures distances as shown.
8 m
10 m
45 m x
What is the distance
across Muddy Pond? 10 8 =
It is 36 meters across Muddy Pond! 45 x
10x = 360
x = 36

10. 3 ways to prove that 2 triangles are similar: AA, SSS, SAS
Summary:
3 ways to prove that 2 triangles are similar:
AA, SSS, SAS

11. Similar Triangles
End of Section 9.3