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7.3 Proving Triangles are Similar

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7.3 Proving Triangles are Similar

Geometry

- Objectives:
- You willuse 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)

- If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar.
- >A ≈ >P & >B ≈ >Q

THEN ∆ABC ~ ∆PQR

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

THEN ∆ABC ~ ∆PQR

AB

BC

CA

=

=

PQ

QR

RP

- 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

XY

If X M and

=

PM

MN

THEN ∆XYZ ~ ∆MNP

∆RST ~ ∆LMN

TR

RS

ST

=

=

LM

NL

MN

- Given:

- Prove

Locate P on RS so that PS = LM. Draw PQ so that PQ ║ RT. Then ∆RST ~ ∆PSQ, by the AA Similarity Postulate, and

RS

ST

TR

=

=

LM

MN

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.

- Which of the three triangles are similar?

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

6

3

CA

12

3

BC

9

3

=

=

=

=

=

=

DE

4

2

FD

8

2

EF

6

2

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

AB

6

CA

12

6

BC

9

=

=

1

=

=

=

GH

6

JG

14

7

HJ

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.

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

Prove: ∆RST ~ ∆PSQ

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

Use the SAS Similarity

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

SR

SP + PR

4 + 12

16

=

4

=

=

=

SP

SP

4

4

ST

SQ + QT

5 + 15

20

=

4

=

=

=

SQ

SQ

5

5

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.

- Ex. 6 – Finding Distance Indirectly.
- To measure the width of a river, you use a surveying technique, as shown in the diagram.

By the AA Similarity Postulate, ∆PQR ~ ∆STR.

RQ

PQ

=

Write the proportion.

RT

ST

RQ

63

=

Substitute.

12

9

RQ

=

12 ● 7

Multiply each side by 12.

RQ

=

84

Solve for TS.

So the river is 84 feet wide.