Unit 4.11 Notes

Proportions in Triangles

Remember the similar triangles that look like this:

Notice: Parallel Lines

Remember the similar triangles that look like this:

Previously, we had to find the length of the small triangle’s sides and the big triangle’s sides to solve for x. But we can actually get

around that step. What do you think our proportion might look like?

The length of TV is 15

The length of TR is x+16

x 516 10=

Side-Splitter TheoremIf a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

ba

dc

XR YS

RQ SQ=

a c

b d=

Do you think the whole shape actually has to be a triangle for this theorem to work?

Nope! It will still work as long as the lines are parallel!

x 5

4 6=

Corollary to the Side-SplitterIf three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

a cb d=

Example

Solve for x and y by using proportions with the lengths of the transversal segments in the middle.

Triangle-Angle-Bisector TheoremIf a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

CD CADB BA=

Example

Solve for x.

Exit Ticket

Solve for x in the three examples above.

1) 2)

3)