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Which Angles Relate to Parallel Lines and Transversals?

parallel lines and transversals: vertical angles

Image credit: Desmos

Parallel lines and transversals are two important geometry concepts because they result in special angle relationships you’ll find in different postulates and theorems and use for solving geometry proofs.

First, two definitions:

Parallel lines: Never intersect, or cross, one another

Transversal line: Intersects two or more lines or line segments

When a transversal line crosses a pair of parallel lines, many types of angles can be created. Let’s go over the many different pairs of angles created by parallel lines and transversals:

Adjacent Supplementary Angles

Supplementary angles are two or more angles that add to 180 degrees. Adjacent angles share a side and a vertex. So, adjacent supplementary angles add to 180 degrees and share a side and vertex.

Try to spot the adjacent supplementary angles:

parallel lines and transversals: supplementary angles

Image credit: Desmos

As you can see, there are many. A and B, B and D, D and C, and C and A are all examples of adjacent supplementary angles. Then, E and F, F and H, and H and G are also adjacent supplementary angles.

Vertical Angles

Vertical angles, sometimes called opposite angles, are opposite each other in two intersecting lines. As the vertical angle theorem says that vertical angles are always congruent angles, the angle measures are the same.

A and D, as well as B and C, are examples of vertical angles in the below diagram. Can you spot some other vertical angle pairs?

parallel lines and transversals: vertical angles

Image credit: Desmos

Answer: E and H as well as F and G are also vertical angle pairs created from these parallel lines and transversal.

Alternate Interior Angles

Interior angles are inside a set of parallel lines. Alternate angles are on the opposite sides of a transversal line.

In the diagram below, C and F are a pair of interior angles, as they’re inside the parallel lines. They’re alternate because they’re on the opposite side of the transversal. When a pair of angles have these two properties, they are considered congruent, alternate interior angles.

Can you spot the other pair of alternate interior angles below?

alternate interior angles

Image credit: Desmos

As you can see, D and E are also alternate interior angles.

Alternate Exterior Angles

Alternate exterior angles are the same as alternate interior angles, except they lie on the outside of two parallel lines.

In the below diagram, A and H are considered alternate exterior angles because they lie on the outside of the two parallel lines and the opposite sides of the transversal line.

Look for other pairs of alternate exterior angles:

alternate exterior angles

Image credit: Desmos

Answer: B and G are also alternate exterior angles.

Angles of Parallel Lines and Transversals

When you see parallel lines and transversals on your geometry worksheets, don’t fear, and remember these key angle characteristics:

  • When a transversal line crosses a pair of parallel lines, you’ll find many pairs of supplementary angles, or angles that add to 180 degrees.
  • Angles directly opposite each other, called vertical angles, are congruent.
  • Alternate interior angles and alternate exterior angles are two other pairs of corresponding angles that are always congruent. Alternate angles are on opposite sides of the transversal, interiors are inside the parallel lines, and exteriors are outside them.

Once familiar with the names for these pairs of angles, you will be well on your way in solving any geometry proof or problem related to parallel lines and transversals.

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