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GeometryLesson 7 – 4
Parallel Lines and Proportional Parts
Objective:Use proportional parts within triangles.
Use proportional parts with parallel lines.
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TheoremTriangle Proportionality Theorem If a line is parallel to one side of a triangle
and intersects the other two sides, then it divides the sides into segments of proportional lengths.
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In triangle PQR ST II RQ. If PT = 7.5, TQ = 3, and SR = 2.5, find PS and PR.
7.5
32.5
5.23
5.7 x
3x = 18.75x = 6.25
PS = 6.25
5.10
5.7
5.2
xx
Method from lastsection:
10.5x = 7.5x + 18.753x = 18.75
x = 6.25PR = 8.75
x
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If PS = 12.5, SR = 5, and PT = 15, find TQ.
12.5
5
15
xx
15
5
5.12
12.5x = 75x = 6
TQ = 6
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TheoremConverse of Triangle Proportionality Theorem If a line intersects two sides of a triangle and
separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.
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In triangle DEF, EH = 3, HF = 9, and DG is one-third the length of GF. Is DE II GH?
3
9
x
3x9
3
3x
x
9x = 9x
The sides are proportional, therefore DE II GH
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DG is half the length of GF, EH = 6, and HF = 10. IS DE II GH?
6
10
x
2x10
6
2x
x
10x = 12x
The lines are not parallel.
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MidsegmentMidsegment of a triangleA segment with endpoints that are the
midpoints of two sides of the triangle.
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TheoremTriangle Midsegment TheoremA midsegment of a triangle is parallel to one side
of the triangle, and its length is one half the length of that side.
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XY and XZ are midsegments of the triangle.
Find XZ
ST
RYXm
6.5 6.5
7
6.5
124o
6.57
14
124
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Find each measure
DE
DB
FEDm
7.5
7.5 9.2
9.282o
7.5
9.2
82
7.5
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Proportional Parts of Parallel Lines
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.
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Frontage is the measurement of a property’s boundary that runs along the side of a particular feature such as a street, lake, ocean, or river. Find the ocean frontage for Lot A to the nearest tenth of a yard.
42
58
60
x
42x = 3480x 82.9
Lot A is approximately 82.9 yards
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CorollaryCongruent Parts of Parallel Lines If three or more parallel lines cut off
congruent segments on one transversal, then they cute off congruent segments on every transversal.
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Find x and y
3y + 8 = 5y - 715 = 2y7.5 = y
6x – 5 = 4x + 32x = 8x = 4
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Find x
34278
8
84
27
xxorx
x
3x = 5x = 5/3
2x + 1 = 3x - 5
6 = x
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Homework
Pg. 489 1 – 9 all, 10 – 26 E, 54 – 66 E