1. 8-5 Proving Triangles Similar

2. WAY BACK WHEN... … ... (in chapter 4)

3. Side - Side - Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

4. USING SIMILARITY THEOREMS
THEOREM Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar. P Q R A B C
If = =
A B PQ BC QR CA RP
then ABC ~ PQR.

5. USING SIMILARITY THEOREMS
THEOREM Side-Angle-Side (SAS) Similarity Theorem X Z Y M P N
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. ZX PM XY MN
If X M and =
then XYZ ~ MNP.

6. Because all of the ratios are equal,  ABC ~  DEF
Using the SSS Similarity Theorem
Which of the following three triangles are similar? A C B 12 6 9 E F D 8 6 4 G J H 14 6 10
SOLUTION
To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides.
Ratios of Side Lengths of  ABC and  DEF
= = , 6 4 AB DE 3 2
Shortest sides
= = , 12 8 CA FD 3 2
Longest sides
= = 9 6 BC EF 3 2
Remaining sides
Because all of the ratios are equal,  ABC ~  DEF

7. Using the SSS Similarity Theorem
Which of the following three triangles are similar? E F D 8 6 4 A C B 12 9 G J H 14 10 A 12 C E F D 8 6 4 G 14 J 6 9 6 10 B H
SOLUTION
To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides.
Ratios of Side Lengths of  ABC and  GHJ
Since  ABC is similar to  DEF and  ABC is not similar to  GHJ,  DEF is not similar to  GHJ.
= = 1, 6 AB GH
Shortest sides
= = , 12 14 CA JG 6 7
Longest sides = 9 10 BC HJ
Remaining sides
Because all of the ratios are not equal,  ABC and  DEF are not similar.

8. Use the given lengths to prove that  RST ~  PSQ.
Using the SAS Similarity Theorem
Use the given lengths to prove that  RST ~  PSQ.
SOLUTION
GIVEN
SP = 4, PR = 12, SQ = 5, QT = 15
PROVE
 RST ~  PSQ P Q S R T
Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides. 12 4 5 15
= = = = 4 SR SP 16 4
SP + PR
= = = = 4 ST SQ 20 5
SQ + QT
The side lengths SR and ST are proportional to the corresponding side lengths of  PSQ.
Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that  RST ~  PSQ.

9. Find the length TS in the enlargement.
Using a Pantograph S R Q T P
10"
2.4"
In the diagram, PR is 10 inches and RT is 10 inches. The length of the cat, RQ, in the original print is 2.4 inches.
Find the length TS in the enlargement.
SOLUTION
Because the triangles are similar, you can set up a proportion to find the length of the cat in the enlarged drawing. = RQ TS PR PT
Write proportion. = 10 20
2.4 TS
Substitute. = TS
4.8
Solve for TS.
So, the length of the cat in the enlarged drawing is 4.8 inches.

10. Use similar triangles to estimate the height of the wall.
Finding Distance Indirectly
ROCK CLIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground.
Similar triangles can be used to find distances that are difficult to measure directly.
Use similar triangles to estimate the height of the wall.
85 ft
6.5 ft
5 ft A B C E D
Not drawn to scale

11. Use similar triangles to estimate the height of the wall.
Finding Distance Indirectly
Use similar triangles to estimate the height of the wall.
SOLUTION
Due to the reflective property of mirrors, you can reason that ACB  ECD.
85 ft
6.5 ft
5 ft A B C E D
Using the fact that  ABC and  EDC are right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar.

12. So, the height of the wall is about 65 feet.
Finding Distance Indirectly
Use similar triangles to estimate the height of the wall.
SOLUTION = EC AC DE BA
Ratios of lengths of corresponding sides are equal.
85 ft
6.5 ft
5 ft A B C E D
So, the height of the wall is about 65 feet. DE 5 = 85
6.5
Substitute.
Multiply each side by 5 and simplify.  DE
65.38