Parallel Lines and Proportional Parts

1
Parallel Lines and Proportional Parts

• By Jacob Begay

2
Theorem 7-4 Triangle Proportionality

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

C
C
C
D
B
B
D
A
A
E
E


3
Theorem 7-5 Converse of the Triangle
Proportionality

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

C
BD
AE
D
B
E
A
4
Theorem 7-6 Triangle Midpoint Proportionality

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

BD ll AE
2BDAE OR BD1/2AE
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Corollary 7-1

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

BC
AC


AF
EF
CD


AE
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Corollary 7-2

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

D
C
BE
CF
GD
B
E
F
G
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Example

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

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.
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Example

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

A
4
3
Q
P
X0.25
3
B
C
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.
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Example

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

Midpoint Segment AB (6,1) Midpoint Segment CB
(7,5)
AC ll DE
AC4
DE4
Therefore 2DE AC