GeometryLesson 7 – 4

Parallel Lines and Proportional Parts

Objective:Use proportional parts within triangles.

Use proportional parts with parallel lines.

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.

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

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

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.

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

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.

MidsegmentMidsegment of a triangleA segment with endpoints that are the

midpoints of two sides of the triangle.

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.

XY and XZ are midsegments of the triangle.

Find XZ

ST

RYXm

6.5 6.5

7

6.5

124o

6.57

14

124

Find each measure

DE

DB

FEDm

7.5

7.5 9.2

9.282o

7.5

9.2

82

7.5

Proportional Parts of Parallel Lines

If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

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

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.

Find x and y

3y + 8 = 5y - 715 = 2y7.5 = y

6x – 5 = 4x + 32x = 8x = 4

Find x

34278

8

84

27

xxorx

x

3x = 5x = 5/3

2x + 1 = 3x - 5

6 = x

Homework

Pg. 489 1 – 9 all, 10 – 26 E, 54 – 66 E