Parallel Lines and Proportional PartsLesson 5-4

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.

Let’s look at the picture!

Triangle Proportionality

BD BE

DA EC

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.

The converse is also a theorem.

Theorem

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

Let’s look at the picture again.

Triangle Proportionality

BD BE

DA ECIf

then the line is parallel to the side of the triangle.

Example 1

AD

B

C

E

In the figure below, CA = 10, CE = 2, DA = 6, BA = 12. Are ED and CB Parallel?

Theorem

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 of the length of the third side.

Here’s the picture.

Theorem

1DE AC

2

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 of the length of the third side.

Why?

Example 2DEF has vertices D(1, 2), E(7, 4), and

F(3,6).a. Find the coordinates of G, the midpoint

of DE, and H, the midpoint of FE.b. What two lines are parallel?c. Compare the lengths of the two parallel

segments.

Corollary 1

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

Here’s the picture.

Corollary 1

a c

b d

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

Corollary 2

If three segments cut off congruent segments on one transversal, then they cut congruent segments of every transversal.

Here’s the picture.

Corollary 2If three segments cut off congruent segments on one transversal, then they cut congruent segments of every transversal.

Example 3You want to plant a row of pine treesalong a slope 80 feet long with thehorizontal spacing of 10 feet, 12 feet, 10feet, 16 feet, 18 feet. How far apart mustyou plant the trees (along the slope?)

Theorem

If two triangles are similar, then the perimeters are proportional to the lengths of corresponding sides.

Theorem

If two triangles are similar, then the lengths of the corresponding altitudes (medians) are proportional to the lengths of the corresponding sides.

Theorem

If two triangles are similar, then the lengths of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides.

Angle Bisector Theorem

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

Here’s the picture.

Angle Bisector Theorem

a d

b e

An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

Why?

Example 4Find x.