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Prove
Given: 𝑙 ∥ 𝑚, 𝑛 and 𝑝 are transversals
Prove: ∠2 is supplementary to ∠8
1
𝑙
𝑚
𝑝2
3 4
5 6
7 8
Solve
6
𝑥=
12
𝑥 + 1
4.2 Similar Triangles
Similar Shapes
• Two shapes are similar when one can become the other after a resize, flip, slide or turn.
• The symbol ~ means “similar to”
Properties:
• Corresponding angles are congruent
• Corresponding side measurements are proportional
Similar ShapesProperties:
• Corresponding angles are congruent
• Corresponding side measurements are proportional
• GIVEN: ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹
Corresponding parts of similar figures
• GIVEN: The triangles are similar.
• ∠A corresponds with ∠____.
• ∠B corresponds with ∠____.
• ∠C corresponds with ∠____.
• AB matches with ______.
• XZ matches with ______.
• BC matches with ______.
A
B C
X
Y
Z30°
30°
70°
70°80°
80°
• AB matches with ______.
• ZY matches with ______.
• BC matches with ______.
• GIVEN: ∆𝐴𝐵𝐶~∆𝑋𝑌𝑍
• ∠A corresponds with ∠____.
• ∠B corresponds with ∠____.
• ∠C corresponds with ∠____.
Corresponding parts of similar figures
A
B C
X
Y
Z
Finding side lengths of similar triangles
• Because side lengths are proportional, we can set up a PROPORTION
L
M
N
RS
T
6
12100
x
Find side lengths 𝐷𝐹 and 𝐸𝐹
𝒙 + 𝟏
B
CA D
E
F
Find side lengths 𝐵𝐶 and 𝐸𝐶
3
𝟒
Angle-Angle Similarity
• If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
A
B C
X
Y Z10°10°
55°
55°
Similarity statement:
Prove if the triangles shown are similar. Write a similarity statement.
A
B C
X
YZ43°
43°
81°
56°
Prove if the triangles shown are similar. Write a similarity statement.
A
B C
X
Y Z
75°
25°
75°
50°
Prove if the triangles shown are similar. Write a similarity statement.
Prove if the triangles shown are similar. Write a similarity statement.
Prove the triangles are similar. Write a similarity statement, and find the side lengths.
𝒙 − 𝟑
Applications of similar triangles
A 40-foot flagpole casts a 25-foot shadow. Find the shadow cast by a nearby building 200 feet tall.
40 ft
25 ft
200 ft
x
Applications of similar triangles
A tree 24 feet tall casts a shadow 12 feet long. John is 6 feet tall. How long is John's shadow? (Draw a diagram and solve)
Applications of similar triangles
A tower casts a shadow 7 m long. A vertical stick casts a shadow 0.6 m long. If the stick is 1.2 m high, how high is the tower? (Draw a diagram and solve)
Applications of similar triangles
A cell phone tower casts a 100-foot shadow. At the same time, a 4-foot 6-inch post near the tower casts a shadow of 3 feet 4 inches. Find the height of the tower.
Applications of similar triangles
A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post?
90 cm 360 cm
