Parallel Lines and Transversals

The triangle proportionality theorem states if a line parallel to one side of a triangle (or transversal) cuts through the other two sides of the triangle, the line divides these two sides proportionally. In the figure below, the transversal DE is parallel to the triangle side BC and intersects both AB and AC, which means it divides them proportionally.

The triangle proportionality theorem also applies if three or more lines are cut by a transversal. Let''s take a closer look.

Intersecting three parallel lines with two transversals & example

In the figure below, two transversal lines intersect three parallel lines. These parallel lines separate the transversals into proportional parts:

In other words, if

AD || BE || CF

, then:

AB/BC = DE/EF

AC/DF = BC/EF

and

AC/BC = DF/EF.

Example

Let''s find the value of x:

First, write a proportion.

GJ/HJ = KL/LM

Substitute the values.

8/6 = 10/x

Use the cross-product.

(8)(x) = (10)(6)

8x = 60

Divide each side by 8.

8x/8 = 60/8

x = 7.5

The value of x is 7.5.

Practice questions on parallel lines and transversals

Answer the following questions using the figure below:

a. If

AB = 5, BC = 6, and DE = 6

, what is the value of

?

First, write a proportion.

AB/BC = DE/EF

Substitute the values.

5/6 = 6/x

Use the cross-product.

(5)(x) = (6)(6)

5x = 36

Divide each side by 5.

5x/5 = 36/5

Answer: x = 7.2

b. If

BC = 3, DE = 5, EF = 6

what is the value of AB?

Write a proportion.

x/BC = DE/EF

Substitute the values.

x/3 = 5/6

Use the cross-product.

(x)(6) = (5)(3)

6x = 15

Divide each side by 6.

6x/6 = 15/6

Answer: x = 2.5

Topics related to the Parallel Lines and Transversals

Parallel Lines

Midpoint Formula

Platonic Solids

Flashcards covering the Parallel Lines and Transversals

Advanced Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Parallel Lines and Transversals

Common Core: High School - Geometry Diagnostic Tests

Advanced Geometry Diagnostic Tests

Learn more about parallel lines and transversals

If your student wants to better understand parallel lines and transversals or needs help with homework or test prep, studying alongside a tutor is the way to go. A qualified private instructor can delve deeper into the triangle proportionality theorem and simplify the process of finding the length of a triangle''s side. Learn more about the benefits of tutoring and how to connect with a private educator by speaking with the Educational Directors at Varsity Tutors today.