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Vertical Angle Theorem: What It Is and How to Use It

The Vertical Angle Theorem says the opposing angles of two intersecting lines must be congruent, or identical in value. That means no matter how or where two straight lines intersect each other, the angles opposite to each other will always be congruent, or equal in value:

Vertical Angle Theorem: Graph showing two straight lines intersect each other

Explaining the Vertical Angle Theorem

According to the Vertical Angle Theorem, when two straight lines intersect, they form two linear pairs. This means that the adjacent angles formed when two lines intersect are supplementary angles, meaning that their angles add up to 180 degrees:

Two straight lines intersect forming adjacent angles

You can see in the perpendicular line figure above that the two lines intersect to form two pairs of vertical angles. Vertical angles are also referred to as vertically opposite angles because they are each on the opposite side of the other. In this figure, angle D and angle B and angles A and C are each a pair of vertically opposite angles. As the Vertical Angle Theorem says, these vertical angles are congruent.

Why We Must Know the Vertical Angle Theorem

As we know, lots of theorems and postulates used in geometry problems explain how angles work such as the Supplementary Angles Theorem, Right Angles Theorem, Angle Addition Postulate, and Triangle Congruence Postulate. Like the rest of these, the Vertical Angles Theorem serves a foundational role in the rules of geometry and trigonometry.

This theorem says that when two straight lines intersect, they form two sets of linear pairs with congruent angles. It also means that the adjacent angles formed by the intersection of these two lines are supplementary, or equal to 180 degrees.

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