chapter 8 lesson 3
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Chapter 8 Lesson 3. Objective: To use AA, SAS and SSS similarity statements. Postulate 8-1 Angle-Angle Similarity (AA ~) Postulate - PowerPoint PPT PresentationTRANSCRIPT
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)