8.5 Proving Triangles are Similar

Geometry Mrs. Spitz Spring 2005

Objectives:

• Use similarity theorems to prove

that two triangles are similar

• Use similar triangles to solve real-

life problems such as finding the height of a climbing wall.

• Assignment: pp.492-493 #1-26

Using Similarity Theorems

• In this lesson, you will study 2

alternate ways of proving that two triangles are similar: Side-Side Side Similarity Theorem and the Side-Angle-Side Similarity Theorem. The first theorem is proved in Example 1 and you are asked to prove the second in Exercise 31.

Side Side Side(SSS) Similarity Theorem

• If the corresponding sides of two

triangles are proportional, then the triangles are similar.

P A B AB PQ = BC QR = CA RP Q R C THEN ∆ABC ~ ∆PQR

X

Side Angle Side Similarity Thm.

• 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.

M P N Z If  X   M and ZX PM = XY MN Y THEN ∆XYZ ~ ∆MNP

Ex. 1: Proof of Theorem 8.2

•Given: •Prove RS LM = ST MN = TR NL ∆RST ~ ∆LMN Locate P on RS so that PS = LM. Draw PQ so that PQ ║ RT. Then ∆RST ~ ∆PSQ, by the AA Similarity Postulate, and RS LM = ST MN = TR NL Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem, it follows that ∆PSQ  ∆LMN Finally, use the definition of congruent triangles and the AA Similarity Postulate to conclude that ∆RST ~ ∆LMN.

Ex. 2: Using the SSS Similarity Thm.

• Which of the three triangles are

similar?

E C 6 14 A 12 4 G 6 9 F 8 D 6 10 B H To decide which, if any, of the triangles are similar, you need to consider the ratios of the lengths of corresponding sides.

Ratios of Side Lengths of ∆ABC and ∆DEF.

J AB DE = 6 4 = 3 2 CA FD = 12 8 = 3 2 BC EF = 9 6 = 3 2  Because all of the ratios are equal, ∆ABC ~ ∆DEF.

Ratios of Side Lengths of ∆ABC

~

∆GHJ

AB GH = 6 6 = 1 CA JG = 12 14 = 6 7 BC HJ = 9 10  Because the ratios are not equal, ∆ABC and ∆GHJ are not similar.

Since ∆ABC is similar to ∆DEF and ∆ABC is not similar to ∆GHJ, ∆DEF is not similar to ∆GHJ.

Ex. 3: Using the SAS Similarity Thm.

• Use the given lengths to prove that

∆RST

~

∆PSQ.

S P 4 5 Q Given: SP=4, PR = 12, SQ = 5, and QT = 15; Prove: ∆RST ~ ∆PSQ R 12 15 Use the SAS Similarity T Theorem. Begin by finding the ratios of the lengths of the corresponding sides.

SR SP = SP + PR SP = 4 + 12 4 = 16 4 = 4

ST SQ = SQ + QT SQ = 5 + 15 5 = 20 5 = 4 So, 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.

Using Similar Triangles in Real Life

• Ex. 6 – Finding Distance indirectly. • To measure the width of a river,

you use a surveying technique, as shown in the diagram.

63 12 9

Solution

63 12 9 By the AA Similarity Postulate, ∆PQR ~ ∆STR.

RQ RT = PQ ST Write the proportion.

RQ 12 = 63 9 Substitute.

RQ = 12 ● 7 Multiply each side by 12.

RQ = 84 Solve for TS.

 So the river is 84 feet wide.