Download 8-3 Proving Triangles Similar

8-3 PROVING TRIANGLES SIMILAR
(p. 432-438)
There is no need to do the Investigation.
When we prove triangles congruent, we no longer use the definition of congruent
triangles since we have learned our shorter methods such as SSS, ASA, SAS, AAS, and
HL.
Likewise, we do not have to always satisfy the definition of similar triangles in order to
prove triangles similar. We are about to learn three shorter methods of proving triangles
similar.
Postulate 8-1 Angle-Angle Similarity (AA ~) Postulate
If two angles of one triangle are congruent to two angles of another triangle, then
the triangles are similar.
Example: Sketch two triangles, ABC and XYZ. If A  X and C  Z, are the
two triangles similar by AA ~?
Example: In the following diagram, BC  AD. Why are the triangles similar? Write a
similarity statement.
B
E
42
42
A
C
D
Theorem 8-1 Side-Angle-Side Similarity (SAS ~) Theorem
If an angle of one triangle is congruent to an angle of a second triangle, and the
sides including the two angles are proportional, then the triangles are similar.
We will not be concerned with the proof of this theorem.
Example: In the following diagram, m B  65 and m K  65. Are these triangles
similar by SAS ~? If so, write a similarity statement.
B
12
65
K
6
65
C
A
20
10
J
L
Theorem 8-2 Side-Side-Side Similarity (SSS ~) Theorem
If the corresponding sides of two triangles are proportional, then the triangles are
similar.
We will not be concerned with the proof of this theorem.
Example: Are the following two triangles similar by SSS ~? If so, write a similarity
statement.
B
15
A
K
21
24
7
5
J
C
8
L
Example: Explain why the following two triangles must be similar. Write a similarity
statement.
Z
18 V 24
12
Y
36
W
X
Do 2 on p. 434.
You can apply these three methods (AA ~, SAS ~, and SSS ~) to find the lengths of
unknown sides in similar triangles. Once you know that two triangles are similar, you
then know that corresponding sides of these two triangles are proportional. This is
sometimes abbreviated as CSSTP. So, you will then set up and solve a proportion to find
an unknown side length.
Example: ABCD is a parallelogram. Why is AWX ~ YWZ? Set up and solve a
proportion to find WY.
A X
5 4W
10
D
Z
B
Y
C
Do 3 on p. 434.
Indirect measurement is a process where you use similar triangles and known lengths to
find distances that are difficult to measure directly. Therefore, you find these new
lengths indirectly.
You can use mirrors or shadows along with similar triangles to obtain lengths or
distances indirectly.
Example: Amber places a mirror 28 feet from the base of a flagpole. When she stands
3 feet from the mirror, she can see the top of the flagpole reflected in the mirror. If her
eyes are 5 ft above the ground, how tall is the flagpole? Make a good sketch. Which
angles are congruent in the two triangles? By what method are the triangles similar? Set
up and solve a proportion to find the height of the flagpole.
Do 4 on p. 435.
Homework p. 435-438: 4,5,7,11,12,15,17,18,23b,24,26,28,38,45,46,53,56,57
15. Solve
2
x

3 x  7.5