Parallel Lines Cut by a Transversal

In this lesson you will expand your understanding of geometric figures through the study of congruency. Congruent figures are figures which are EXACTLY the same. They are the same shape (angle measures are equal) and the same size (side lengths are equal). In order to know if figures are congruent, it is helpful to know whether angles within the figures are congruent.

Congruent means same shape and same size.

We will begin with a review of parallel lines. You will see parallel lines in later lessons and it will be important to know where your congruent angles are.

image of a set of parallel lines and an intersecting line

Congruent Angle Relationships

Angle Relationship

Description

Examples

Vertical Angles

(Congruent)

When two lines cross each other, angles which are diagonally across the intersection from each other are called Vertical Angles. Vertical Angles are congruent.

∠A&∠D

∠B&∠C

∠E&∠H

∠F&∠G

Corresponding Angle

(Congruent)

Angles which are in the same location relative to the transversal's intersection with the parallel lines. Corresponding angles are congruent.

∠A&∠E

∠C&∠G

∠B&∠F

∠D&∠H

Alternate Interior Angles

(Congruent)

Angles which are on opposite sides of the transversal (Alternate) and in between the parallel lines (interior). Alternate interior angles are congruent.

∠D&∠E

∠F&∠C

Alternate Exterior Angles

(Congruent)

Angles which are on opposite sides of the transversal (Alternate) and outside the parallel lines (exterior). Alternate exterior angles are congruent.

∠A&∠H

∠B&∠G

While each of these angle pair relationships result in congruent angles, the most commonly used relationships for this module will be Vertical Angles and Alternate Interior Angles.

image of a set of parallel lines and an intersecting line

Supplementary Angle Relationships

Angle Relationship

Description

Examples

Linear Pair

(supplementary)

Two adjacent angles which together form a (straight) line or segment. The two angles will sum to 180 degrees.

∠A&∠B

∠A&∠C

∠B&∠D

∠D&∠C

∠E&∠F

∠E&∠G

∠G&∠H

∠H&∠F

Same Side Interior

(supplementary)

Angles which are on the same side of the transversal and in-between the parallel lines. Same side interior angles sum to 180.

∠C&∠E

∠F&∠D

Same Side Exterior

(supplementary)

Angles which are on the same side of the transversal and outside the parallel lines. Same side exterior angles sum to 180.

∠A&∠G

∠B&∠H

Each of these angle pairs result in supplementary angles and will therefore yield a sum of 180 degrees.

Knowing the relationships listed above is the first step in building your mathematical arguments (Proofs). Check out the video below to see a demonstration of these angle relationships.

Test your knowledge of angle relationships by completing the matching activity below.

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