1. Indirect Measurement
and Additional
Similarity Theorems 8.5

2. Learn the triangle angle bisector theorem.
Learn the proportional altitudes theorem.
Learn the proportional medians theorem.
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3. Proportional Angle Bisectors Theorem
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4. Proportional Altitudes Theorem
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5. Proportional Medians Theorem
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6. Proportional Angle Bisector Theorem
Find PS and SR.
40(x – 2) = 32(x + 5)
PS = x – 2
SR = x + 5
40x – 80 = 32x + 160
PS = 28
SR = 35
8x = 240
x = 30
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7. Proportional Altitudes Theorem
The drawing of the table below has legs AE and CG. AC measures 12 inches and GE measures 36 inches. If BD measures 7 inches, what is the measure of DF and what is the height of the table? A B C D
12x = 252
x = 21
Therefore, DF = 21, and the table is 28 inches tall.
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8. Proportional Medians Theorem
In the figure, EFD ~ JKI. EG is a median of EFG and JL is a median of JKL. Find JL if EF = 36, EG = 18, and JK = 56. 36 18 56 E D G F K J I L x
1008 = 36x
x = 28
Therefore, JL = 28.
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9. Use indirect measure to find the missing value.
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10. Use indirect measure to find the missing value.
10x = 360
x = 36
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11. Use indirect measure to find the missing value.
x = 27 feet
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12. Use indirect measure to find the missing value.
4x = 55
x = feet
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13. Use the Triangle Angle Bisector Theorem find ST and SR
7.5x = 12.5x – 62.5
–5x = –100
x = 20
SR = 25, ST = 15
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14. EXAMPLE 4
In the diagram, QPR  RPS. Use the given side lengths to find the length of RS .
7x = 195 – 13x
x = 9.75
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15. Use the Triangle Angle Bisector Theorem find x.
42x2 = 50x2 – 20x
8x2 – 20x = 0
4x(2x – 5) = 0
x = 0 or x = 2.5
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16. Use the Triangle Angle Bisector Theorem find x.
4x = 36 – 6x
10x = 36
x = 3.6
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17. Assignment
8.5 Indirect Measurment