7 3 proving triangles are similar
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7.3 Proving Triangles are Similar. Geometry. Objectives/DFA/HW. Objectives: 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

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7.3 Proving Triangles are Similar

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7 3 proving triangles are similar

7.3 Proving Triangles are Similar

Geometry


Objectives dfa hw

Objectives/DFA/HW

  • 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)


Angle angle aa similarity postulate

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

THEN ∆ABC ~ ∆PQR


Side side side sss similarity theorem

Side Side Side(SSS) Similarity Theorem

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

THEN ∆ABC ~ ∆PQR

AB

BC

CA

=

=

PQ

QR

RP


Side angle side similarity theorem

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.

ZX

XY

If X  M and

=

PM

MN

THEN ∆XYZ ~ ∆MNP


Ex 1 proof of theorem 8 2

Ex. 1: Proof of Theorem 8.2

∆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.


Ex 2 using the sss similarity theorem

Ex. 2: Using the SSS Similarity Theorem.

  • 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.


Ratios of side lengths of abc ghj

Ratios of Side Lengths of ∆ABC ~ ∆GHJ

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.


Ex 3 using the sas similarity theorem

Ex. 3: Using the SAS Similarity Theorem.

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


7 3 proving triangles are similar

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.


Using similar triangles in real life

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.


Solution

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

Solution

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.


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