Bell Ringer

Proving Triangles are Similar by AA,SS, & SAS

Use the AA Similarity PostulateExample 1

Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning.SOLUTION

If two pairs of angles are congruent, then the triangles are similar.1. G L because they are both marked as right angles.

mF + 90° + 61° = 180° Triangle Sum Theorem

mF + 151° = 180° Add.

mF = 29° Subtract 151° from each side.

Use the AA Similarity PostulateExample 2

Are you given enough information to show that RST is similar to RUV? Explain your reasoning.SOLUTION

Redraw the diagram as two triangles: RUV and RST.

From the diagram, you know that both RST and RUV measure 48°, so RST RUV. Also, R R by the Reflexive Property of Congruence. By the AA Similarity Postulate, RST ~ RUV.

Now You Try Use the AA Similarity Postulate

Determine whether the triangles are similar. If they are similar, write a similarity statement.

1.

2.

ANSWER yes; RST ~ MNL

ANSWER yes; GLH ~ GKJ

Use Similar TrianglesExample 3

A hockey player passes the puck to a teammate by bouncing the puck off the wall of the rink, as shown below. According to the laws of physics, the angles that the path of the puck makes with the wall arecongruent. How far from the wall will the teammate pick up the pass?

AB

DE=

BC

ECWrite a proportion.

25

x=

40

28 Substitute x for DE, 25 for AB, 28 for EC, and 40 for BC.

x · 40 = 25 · 28 Cross product property

Use Similar TrianglesExample 3

40x = 700 Multiply.

40

40x=

40

700Divide each side by 40.

x = 17.5 Simplify.

The teammate will pick up the pass 17.5 feet from the wall.

ANSWER

Checkpoint Use Similar Triangles

Write a similarity statement for the triangles. Then find the value of the variable.

3.

4.

ANSWER ABC ~ DEF; 9

ABD ~ EBC; 3ANSWER

Now You Try

Example 1 Use the SSS Similarity Theorem

Find the ratios of the corresponding sides.

SOLUTION

Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A.

All three ratios are equal. So, the corresponding sides of the triangles are proportional.

PRSU

126

12 ÷ 66 ÷ 6= = 2

1=

RQUT

105

10 ÷ 55 ÷ 5= = 2

1=

QPTS

84

8 ÷ 44 ÷ 4= = 2

1=

Example 1 Use the SSS Similarity Theorem

ANSWER

The scale factor of Triangle B to Triangle A is .21

By the SSS Similarity Theorem, PQR ~ STU.

Example 2 Use the SSS Similarity Theorem

Is either DEF or GHJ similar to ABC?

SOLUTION

Look at the ratios of corresponding sides in ABC and DEF.1.

Shortest sides

= 64

ABDE

32=

Longest sides

= 128

CAFD

32=

Remaining sides

= 96

BCEF

32=

ANSWERBecause all of the ratios are equal, ABC ~ DEF.

Example 2 Use the SSS Similarity Theorem

Look at the ratios of corresponding sides in ABC and GHJ.2.

Shortest sides

=66

AB

GH

11

=

Longest sides

=1214

CA

JG

67

=

Remaining sides

BC

HJ

910

=

ANSWERBecause the ratios are not equal, ABC and GHJ are not similar.

Is either DEF or GHJ similar to ABC?

Checkpoint Use the SSS Similarity Theorem

ANSWER yes; ABC ~ DFE

ANSWER no

Determine whether the triangles are similar. If they are similar, write a similarity statement.

1.

2.

Now You Try

Example 3 Use the SAS Similarity Theorem

Determine whether the triangles are similar. If they are similar, write a similarity statement.

SOLUTION

C and F both measure 61°, so C F.Compare the ratios of the side lengths that include C and F.

=ACDF

35

Shorter sides = 610

CBFE

35=Longer sides

The lengths of the sides that include C and F are proportional.

ANSWERBy the SAS Similarity Theorem,

ABC ~ DEF.

Example 4 Similarity in Overlapping Triangles

Show that VYZ ~ VWX.

Separate the triangles, VYZ and VWX, and label the side lengths.

SOLUTION

V V by the Reflexive Property of Congruence.

Shorter sides

4 + 84=VY

VW124= 3

1=

Longer sides

5 + 105=ZV

XV155= 3

1=

Example 4 Similarity in Overlapping Triangles

ANSWERBy the SAS Similarity Theorem, VYZ ~ VWX.

The lengths of the sides that include V are proportional.

Checkpoint Use the SAS Similarity Theorem

Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning.

3.

ANSWER No; H M but 68

8

12≠ .

Now You Try

Checkpoint Use the SAS Similarity Theorem

Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning.

4.

ANSWER

Yes; P P, and the lengths of the sides that include P are proportional,

so PQR ~ PST by the SAS Similarity Theorem.

,63

PSPQ

= = 21

105

PTPR

= = 21 ;

Now You Try

Page 375 #s 8-26 &

Page 382 #s 6-18 even only