Chapter 8 Lesson 3

Objective: To use AA, SAS and SSS similarity

statements.

Postulate 8-1  Angle-Angle Similarity (AA ~) Postulate

                                                                                   If two angles of one triangle are congruent to

two angles of another triangle, then the triangles are similar.

∆TRS ~ ∆PLM

RSW        VSB because vertical angles are congruent.  R     V because their measures are equal. ∆RSW ~ ∆VSB by the Angle-Angle Similarity Postulate.

Example 1: Using the AA~ Postulate

Explain why the triangles are similar. Write a similarity statement.

                                                                                                

Explain why the triangles are similar. Write a similarity statement.

                                                                      

                          

Example 2: Using the AA~ Postulate

ABMX

B X A

K

M

58°58° 58°58°

∆AMX~∆BKX by AA~ Postulate

Theorem 8-1  Side-Angle-Side Similarity (SAS ~)

TheoremIf an angle of one triangle is congruent to an

angle of a second triangle, and the sides including the two angles are proportional,

then the triangles are similar.

Theorem 8-2  Side-Side-Side Similarity (SSS ~)

TheoremIf the corresponding sides of two triangles

are proportional, then the triangles are similar.

Example 3: Using Similarity Theorems

                                                                                 

   Explain why the triangles must be similar. Write a similarity statement.

QRP     XYZ because they are right angles.

Therefore, ∆QRP ~ ∆XYZ by the SAS ~ Theorem

43

86

43

ZYPR

andXYOR

Example 4: Using Similarity Theorems

Explain why the triangles must be similar. Write a similarity statement.                 

43

EFAB

GFCB

EGAC

∆∆ABC ~ ∆EFG by SSS~ ABC ~ ∆EFG by SSS~ TheoremTheorem

Example 5: Finding Lengths in Similar Triangles

  Explain why the triangles are similar. Explain why the triangles are similar. Write a similarity statement. Write a similarity statement.

Then find Then find DEDE. . EBDABC

Because vertical angles are congruent.

EBAB

32

1812

DBCB

32

2416

ΔABC ~ ΔEBD by the SAS~ Theorem.

32

DECA

3210

DE

302 DE15DE

Example 6: Finding Lengths in Similar Triangles

Find the value of Find the value of xx in the figure. in the figure.

                                   

1286

x

x872

x9

Indirect Measurement is when you use similar triangles and measurements to find distances that are difficult to measure directly.

Example 7: Indirect MeasurementGeologyGeology Ramon places a mirror on the ground 40.5 ft from Ramon places a mirror on the ground 40.5 ft from the base of a geyser. He walks backwards until he can see the base of a geyser. He walks backwards until he can see the top of the geyser in the middle of the mirror. At that the top of the geyser in the middle of the mirror. At that point, Ramon's eyes are 6 ft above the ground and he is 7 point, Ramon's eyes are 6 ft above the ground and he is 7 ft from the image in the mirror. Use similar triangles to find ft from the image in the mirror. Use similar triangles to find the height of the geyser. the height of the geyser.

∆HTV ~ ∆JSV

SVTV

JSHT

5.4076

x

x7243 x7.34

The geyser is about 35 ft. high.

Example 8: Indirect Measurement

In sunlight, a cactus casts a 9-ft shadow. At the same In sunlight, a cactus casts a 9-ft shadow. At the same time a person 6 ft tall casts a 4-ft shadow. Use similar time a person 6 ft tall casts a 4-ft shadow. Use similar triangles to find the height of the cactus. triangles to find the height of the cactus.

XX

99

66

44

649 x

x454 x5.13

AssignmentAssignment

Page 437 #1-39(odd)