7.3 Proving Triangles are Similar

Geometry

Objectives/DFA/HWObjectives:

You will use similarity theorems to prove that two triangles are similar.

You will use similar triangles to solve real-life problems such as finding the height of a climbing wall.

DFA: p.456 #24

HW: pp.455-457 (2-28 even)

Angle-Angle (AA~) Similarity Postulate

If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.

>A ≈ >P & >B ≈ >Q

A

B C

P

Q R

THEN ∆ABC ~ ∆PQR

Side Side Side(SSS) Similarity Theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

A

B C

P

Q R

AB

PQ QR RP

BC CA= =

THEN ∆ABC ~ ∆PQR

Side Angle Side Similarity Theorem. 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.

X

Z Y

M

P N

If X M andZX

PM=

XY

MN

THEN ∆XYZ ~ ∆MNP

Ex. 1: Proof of Theorem 8.2

Given: ProveRS

LM MN

NL

ST TR= =

∆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 MN

NL

ST TR= =

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 Theorem. Which of the three triangles are similar?

96

12A

B

C 6 4

8D

E

F106

14G

H

J

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.

AB

DE 4 2

6 3= =

CA

FD 8 2

12 3= =

BC

EF 6 2

9 3= =

Because all of the ratios are equal, ∆ABC ~ ∆DEF.

Ratios of Side Lengths of ∆ABC ~ ∆GHJ

AB

GH 61

6= =

CA

JG 14 7

12 6= =

BC

HJ 10

9=

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 Theorem. Use the given lengths to prove that ∆RST ~ ∆PSQ.

1512

54

S

R T

P Q

Given: SP=4, PR = 12, SQ = 5, and QT = 15;

Prove: ∆RST ~ ∆PSQ

Use the SAS Similarity

Theorem. Begin by finding the ratios of the lengths of the corresponding sides.

SRSP

SP + PRSP

4 + 124

= = =164

= 4

STSQ

SQ + QTSQ

5 + 155

= = =205

= 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.

9

12

63

9

12

63

SolutionBy the AA Similarity Postulate, ∆PQR ~ ∆STR.RQ

RT ST

PQ=

RQ

12 9

63=

RQ 12 ● 7=

Write the proportion.

Substitute.

Solve for TS.RQ 84=

Multiply each side by 12.

So the river is 84 feet wide.