Parallel Lines and Proportional Parts

Section 7.4

Proportional parts of triangles

• Non parallel transversals that intersect parallel lines can be extended to form similar triangles.

• So, the sides of the triangles are proportional.

Side Splitter Theorem or Triangle Proportionality Theorem

• If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.

• If BE || CD, then

•

A

C D

B EAB AE

BC ED

AB||ED, BD = 8, DC = 4, and AE = 12.

• Find EC

• by ?

• EC = 6

C

B A

D E

CD CE

DB EA

4

8 12

EC

UY|| VX, UV = 3, UW = 18, XW = 16.

• Find YX.

• YX = 3.2

W

U

YX

VUV YX

WV XW

3

15 16

YX

Converse of Side Splitter

• If a line intersects the other two sides and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.

• If

• then BE || CD

A

C D

B E

AB AE

BC ED

Determine if GH || FE. Justify

• In triangle DEF, DH = 18, and HE = 36, and DG = ½ GF.

• To show GH || FE,

• Show

• Let GF = x, then DG = ½ x.

DG DH

GF HE

• Substitute

• Simplify

• Simplify

118236

x

x

1182

1 36

1 1

2 2

• Since the sides are proportional, then GH || FE.

Triangle Midsegment Theorem

• A midsegment of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side.

• If D and E are mid-

• Points of AB and AC,

• Then DE || BC and

• DE = ½ BC

Example

• Triangle ABC has vertices A(-2,2), B(2, 4) and C(4,-4). DE is the midsegment of triangle ABC.

• Find the coordinates of D and E.

• D midpt of AB• D(0,3)•

2 2 2 4,

2 2

• E midpt of AC

• E(1, -1)

• Part 2 - Verify BC || DE

• Do this by finding slopes

• Slope of BC = -4 and slope of DE = -4

• BC || DE

• Part 3 – Verify DE = ½ BC

• To do this use the distance formula

• BC = which simplifies to

• DE =

• DE = ½ BC

68 2 17

17

Corollaries of side splitter thm.

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

• If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Examples

• 1. In the figure, Larch, Maple, and Nutthatch Streets are all parallel. The figure shows the distance between city blocks. Find x.

• Find x and y.

• Given: AB = BC

• 3x + 4 = 6 – 2x

• X = 2

• Use the 2nd corollary to say DE = EF

• 3y = 5/3 y + 1

• Y = ¾