section 6 – 6 use proportionality theorem. theorems triangle proportionality theorem – if a line...
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Section 6 – 6
Use Proportionality Theorem
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TheoremsTriangle Proportionality Theorem
– If a line parallel to one side of a triangle intersects the other two sides, then it divided the two sides proportionally.
Converse of the Triangle Proportionality Theorem –
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
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TheoremsTheorem 6.6 – If three parallel lines
intersect two transversals, then they divide the transversals proportionally.
Theorem 6.7 – If a ray bisects an angle of a triangle, then
it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
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Example 1In the diagram, RS || PN, MS = 15, SN = 20, and RP = 12. What is the length of MR?
MS = MR SN RP
MR 12
Triangle Proportionality Theorem
15 = 20
Cross Multiply
20MR = 180
S
N
M
P
12
15
R
20 20
20
MR = 9
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Example 2In the diagram, ABD = CBD. Use the given side lengths to find the length of DC.
Because BD is an angle bisector of ABC, we can apply Theorem 6.6.
DA = DC
BABCD
3224
40
x C
B
A
Angle bisector divides opposite side proportionally.
Substitute 40 – x = x
24 32
24x =
32(40 – x)24x = 1280 – 32x
~
56x = 1280 x = 22.9
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HomeworkSection 6-6
Page 400 – 403 3 – 6, 8 – 11,
13 – 16, 30 – 33