Alternate Interior Angles (Definition, Theorem, & Examples)

  1. Video
  2. Key Terms
  3. Alternate Interior Angles Definition
  4. Alternate Interior Angles Theorem
  5. Alternate Interior Angles Examples
  6. Alternate Interior Angles In Real Life

Key Terms

Sometimes geometry feels like a giant parts warehouse. You trade a lot of number-crunching (not much addition, multiplication, subtraction or division in geometry) for a lot of inventory.

Parts of an Angle

For example, let's construct angle Z. We almost never write "angle Z," using instead a quick shorthand, Z. Something as simple as an angle has parts.

Alternate Interior Angles - Parts of an angle, transversal, parallel lines

  1. Two rays, ZA and ZU, meet at Point Z
  2. Where they meet at Point Z, they form a vertex, Z

We say rays ZA and ZU, but those rays could also be small snippets out of longer lines that intersected at Point Z. They could be snippets cut as rays or as line segments, depending on taking an infinite chunk or a finite chunk of the infinite, intersecting lines.

Parallel Lines

Unlike the intersecting rays ZA and ZU, parallel lines never meet. The two lines, line segments, or rays never converge (move closer) or diverge (move away). The only sneaky way to get an angle from parallel lines is to declare each line is a straight angle, with a measure of 180°. While two points determine a line, if you locate three points on a line, you have created a straight angle with the middle point as the vertex.

Transversals

Parallel lines can be intersected by transversals. Any line cutting across parallel lines is a transversal. It can cross at any angle. A transversal intersecting parallel lines at 90° is perpendicular.

Alternate Interior Angles Definition

When a transversal intersects parallel lines, it creates an interior and exterior. Think about it: if you were two-dimensional and came across a line in your path, that line would stretch infinitely in two directions and you could not get past it. You would be outside, at the exterior, of the parallel lines. Just beyond the line and between it and the parallel line next to it, is the interior.

When the transversal intersects, it creates four angles at each parallel line, or eight angles altogether. Four of those angles are exterior and four are interior. We are interested in the four interior lines, those are our Alternate Interior Angles.

Let's create parallel lines LI and ON, and a transversal HE.

Alternate Interior Angles

The two points where HE crosses the parallel lines are Points A and R. Yes, we have a HARE crossing a LION.

You mark drawings of parallel lines with little bird-feet marks, like Vs on their sides. Notice we have four exterior angles:

  1. HAL
  2. HAI
  3. ORE
  4. NRE

We have four interior angles:

  1. LAR
  2. IAR
  3. ARO
  4. ARN

We are only interested in the four interior angles. Two of the interior angles are built using the top parallel line LI, and two are built using the bottom parallel line, ON.

Alternate interior angles are two congruent angles from different parallel lines (one from LI, one from ON). This says:

Alternate Interior Angles Example

  • LAR is an alternate interior angle with ARN
  • IAR is an alternate interior angle with ARO.

Alternate Interior Angles Theorem

Once you can recognize and break apart the various parts of parallel lines with transversals, you can use the Alternate Interior Angles Theorem to speed up your work.

Alternate Interior Angles Theorem

The Alternate Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Converse of Alternate Interior Angles Theorem

The converse of the theorem is also true:

The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Alternate Interior Angles Examples

We can prove both these theorems so you can add them to your toolbox. With our origional figure, LI and ON are parallel lines (given) transversed by HE (given).

Alternate Interior Angles

We could declare all sorts of relationships, but the proof can be short and simple:

Using the alternate Interior Angles Theorem

  • LI  ON (given)
  • LAR  ORE (Corresponding Angles Postulate)
  • ORE  ARN (Vertical Angles Theorem)
  • LAR  ARN (Transitive Property of Congruence)

The Transitive Property of Congruence says if A is like B and B is like C, then A is like C. Since LAR was congruent to ORE, and ORE was congruent to ARN, then:

The Transitive Property of Congruence for alternate interior angles

  • LAR  ORE  ARN
  • LAR  ARN

To prove the converse, "If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel," we work the other way around:

  • LI and ON with transversal HE (given)
  • LAR  ARN (given)
  • LAR  HAI (Vertical Angles Theorem)
  • HAI  ARN (Transitive Property of Congruence)
  • LI  ON (Converse of Corresponding Angles Theorem)
The Converse of the Corresponding Angles Theorem says that if two lines and a transversal form congruent corresponding angles, then the lines are parallel.

Alternate Interior Angles in Real Life

Look at a window with panes divided by muntins. The parallel, vertical muntins are probably transversed by a horizontal muntin. Anywhere they cross, you can find alternate interior angles.

Alternate Interior Angles

Make a capital letter Z, like the Mark of Zorro (you'll probably have to look up that 1919 superhero). The top and bottom horizontal slashes of Zorro's sword are parallel lines, and the diagonal slash is the transversal. Zorro's big Z makes two obvious, alternate interior angles.

Alternate Interior Angles

Lesson Summary

After navigating this lesson, you are now able to define the parts of an angle (lines, rays or line segments meeting at an endpoint and forming a vertex). You can also draw, describe and identify transversal lines, straight lines, straight angles, parallel lines, and alternate interior angles.

In addition, you can now apply the Alternate Interior Angles Theorem to find angles in parallel lines crossed by a transversal. You definitely figured out the angles on this one!

Next Lesson:

Transversal Liness & Angles

What you learned:

After working your way through this lesson and video, you will learn to:

  • Define the parts of an angle
  • Draw, describe and identify parallel lines, straight lines, straight angles, transversals, and alternate interior angles
  • Apply the Alternate Interior Angles Theorem and its converse to find angles in parallel lines crossed by a transversal, and to prove lines parallel
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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Ashburn, VA

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