* Parallel Lines and Proportional Parts

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

Example 1 In the figure below, CA = 10, CE = 2, DA = 6, BA = 12. Are ED and CB Parallel? C E A B D

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 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 2 DEF has vertices D(1, 2), E(7, 4), and F(3,6). Find the coordinates of G, the midpoint of DE, and H, the midpoint of FE. What two lines are parallel? 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 If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

Corollary 2 If three parallel segments cut off congruent segments on one transversal, then they cut congruent segments of every transversal. Here’s the picture.

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

Example 3 You want to plant a row of pine trees along a slope 80 feet long with the horizontal spacing of 10 feet, 12 feet, 10 feet, 16 feet, 18 feet. How far apart must you 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 An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.

Why?

Example 4 Find x.