In z-pattern. Notice that the pair of alternate

In this lesson, you will learn how to identify alternate interior angles and how to use the theorem to find missing angles and to solve everyday geometry problems.

Definitions

An angle is formed when two rays, a line with one endpoint, meet at one point called a vertex. The angle is formed by the distance between the two rays. Angles in geometry are often referred to using the angle symbol so angle A would be written as angle A.

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Angle

A transversal line is a line that crosses or passes through two other lines. Sometimes, the two other lines are parallel, and the transversal passes through both lines at the same angle.

The two other lines don’t necessarily have to be parallel in order for a transversal to cross them.A straight angle, also called a flat angle, is formed by a straight line. The measure of this angle is 180 degrees. A straight angle can also be formed by two or more angles that sum to 180 degrees. Here, angle 1 + angle 2 = 180.

Transversal Line

Parallel lines are two lines on a two-dimensional plane that never meet or cross.

When a transversal passes through parallel lines, there are special properties about the angles that are formed that do not occur when the lines are not parallel. Notice the arrows on lines m and n towards the left. These arrows indicate that lines m and n are parallel.

parallel lines

Alternate interior angles are formed when a transversal passes through two lines. The angles that are formed on opposite sides of the transversal and inside the two lines are alternate interior angles. Notice the pairs of blue and pink angles.

Alternate Interior Angles

These pairs are alternate interior angles.

Another way to think about alternate interior angles is by using the z-pattern. Notice that the pair of alternate interior angles makes a Z.

z pattern

In this window pane, angle a and angle b are alternate interior angles because they are on opposite sides of the transversal but inside the parallel lines.

Window Pane

Alternate Interior Angles Theorem

The Alternate Interior Angles theorem states, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.A theorem is a proven statement or an accepted idea that has been shown to be true. The converse of this theorem, which is basically the opposite, is also a proven statement: if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.These theorems can be used to solve problems in geometry and to find missing information. This diagram shows which pairs of angles are equal and alternate interior.

Notice that the lines are parallel.

alternate interior angles

Examples

1.

Use the Alternate Interior Angle theorem to find the measure of angle x, angle y, and angle z. Assume the lines are parallel.

alternate interior angles example

First, we need to identify a pair of alternate interior angles.

Angle y and 58 are on opposite sides of the transversal and inside the parallel lines, so they must be alternate interior. Since the lines are parallel, angle y = 58.Next, notice that angle x and 58 form a straight angle. Since a straight angle measures 180 degrees, angle x + 58 = 180 and 180 – 58 = angle x, so angle x = 122.Finally, angle x and angle z are alternate interior angles, and we know that alternate interior angles are equal.

So, angle x = 122 then angle z = 122.2. Use the Alternate Interior Angles theorem to first find x and y and then to find the measures of the angles. Assume the lines are parallel.

ala example 2

First, we need to identify a pair of alternate interior angles. 2x + 13 and 3x – 24 are on opposite sides of the transversal and inside the parallel lines, so they must be alternate interior.

Since the lines are parallel, they are equal, so 2x + 13 = 3x – 24. Now, gather terms to find x. 37 = x.Next, plug x into 2x + 13 and 3x – 24 to find the measure of the angles.

2(37) + 13 = 87 and 3(37) – 24 = 87.Now since 2x + 13 and 3y + 24 form a straight angle, their sum is 180 degrees. Thus, (2x + 13) + (3y + 24) = 180.

Since 2x + 13 = 87, we can substitute 87 in for 2x + 13 in the equation to get 87 + 3y + 24 = 180.Collect terms to solve for y. 3y + 111 = 180. Subtract 111 from both sides, and you get 3y = 69.

Divide by 3 on both sides, and y = 23.Now that we know the value of y, plug it back into 3y + 24 to find the angle measure: 3(23) + 24 = 93.

Lesson Summary

Alternate interior angles are formed when a transversal passes through two lines. The angles that are formed on opposite sides of the transversal and inside the two lines are alternate interior angles.

The theorem says that when the lines are parallel, that the alternate interior angles are equal. This theorem can be used to solve problems in geometry.

Key Terms

Straight Angle
Straight Angle

Angle – the space between two rays which meet at a vertexRay – a line with one endpointVertex – the meeting point for two raysTransversal line – a line that crosses or passes through two other linesStraight angle – an angle formed by a straight line; also known as a flat angleParallel lines – two lines on a two-dimensional plane that never meet or crossAlternate interior angles – formed when a transversal passes through two linesAlternate Interior Angles theorem – theorem which states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruentTheorem – a proven statement or an accepted idea that has been shown to be trueConverse – opposite; often used in the context of theorems

Learning Outcomes

After this lesson, you should be able to:

  • Define the parts of an angle
  • Describe and/or illustrate transversal lines, straight lines, parallel lines, and alternate interior angles
  • Apply the Alternate Interior Angles theorem to find angles given a set of parallel lines with a transversal line
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