Parallel Lines and Parallel Lines and Proportional PartsProportional Parts

By: Jacob BegayBy: Jacob Begay

Theorem 7-4 Triangle Theorem 7-4 Triangle Proportionality:Proportionality:

If a line is parallel to one side of a If a line is parallel to one side of a triangle and intersects the other two triangle and intersects the other two sides in two distinct points, then it sides in two distinct points, then it separates these sides into segments separates these sides into segments of proportional lengths. of proportional lengths.

A

B

C

D

EB

C

DA

C

ECB

CA

CD

CE=

BD

AE=

Theorem 7-5 Converse of the Theorem 7-5 Converse of the Triangle Proportionality:Triangle Proportionality:

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

A

B

C

D

E

BD AE

Theorem 7-6 Triangle Midpoint Theorem 7-6 Triangle Midpoint Proportionality:Proportionality:

A segment whose endpoints are the A segment whose endpoints are the midpoints of two sides of a triangle is midpoints of two sides of a triangle is parallel to the third side of the parallel to the third side of the triangle, and its length is one-half the triangle, and its length is one-half the length of the third side. length of the third side.

A

B

C

D

E 2BD=AE OR BD=1/2AE

BD ll AE

Corollary 7-1Corollary 7-1

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

A

BC

D

EF G

BCEF

CDFG

ABAE

ADAG

=

=

ACAF

BCEF

CDAE

FGAB

=

=

Corollary 7-2Corollary 7-2

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

BC

D

E F G

BE CF GD

ExampleExample

Based on the figure below, which Based on the figure below, which statement is false?statement is false?

A

D

B C

E

3 4

34

A.DE is Parallel to BC C.ABC ~ ADE

B.D is the Midpoint of AB D.ABC is congruent to ADE

D. ABC is congruent to ADE. Corresponding sides of the triangles are proportional but not congruent.

ExampleExample

Find the value of X so that PQ is Find the value of X so that PQ is parallel to BC.parallel to BC.

A

P Q

B C

3 4

3X+0.25

A.1 C.1.25

B.2.5 D.2

D. 2 Since the corresponding segments must be proportional for PQ to be parallel to BC.

ExampleExample Triangle ABC has vertices A (0,2), B (12,0), and C (2,10). Triangle ABC has vertices A (0,2), B (12,0), and C (2,10). A. Find the coordinates of D, the midpoint of Segment AB, and E, the A. Find the coordinates of D, the midpoint of Segment AB, and E, the

midpoint of Segment CB.midpoint of Segment CB. B. Show that DE ll AC.B. Show that DE ll AC. C. Show that 2DE = AC.C. Show that 2DE = AC.

D0+12, 2+0

2 2= Or D = (6,1)

E12+2, 0+10

2 2= Or E = (7,5)

Midpoint Segment AB (6,1)

Midpoint Segment CB (7,5)

Slope of AC = 2-10

0-2AC=4

Slope of DE = 1-5

6-7DE=4AC ll DE

AC= (0-2) + (2-10) 2 2

= 4+64

= 68 Or 2 17

DE = (6-7) + (1-5)2 2

= 1+16 Or 17

Therefore 2DE = AC