Alternate Interior Angles Theorem


The alternate interior angles theorem states that if a transversal intersects or cuts two lines that are parallel, then the alternate interior angles are congruent or equal.

In the figure below, the red line is the transversal or the line that cuts or intersects the two blue lines that are parallel. Therefore, the alternate interior angles shown in green, named 1 and 2, are congruent or equal.

Alternate interior angles theorem

Examples showing how to use the alternate interior angles theorem


1.

Use the figure below to find the measure of angle x.

Alternate interior angles theorem example

Since line p is parallel to line q, angle x has the same value as the angle whose value is 75 degrees.

x = 75 degrees.

2.

Use the figure below to find the value of a.

Alternate interior angles theorem example

Since line p is parallel to line q, (2a + 30)0 has the same value as the angle whose value is 145 degrees.

2a + 30 = 140

2a + 30 - 30 = 140 - 30

2a = 110

a = 55

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