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Vertical Angles                       Page 2 of 3

Example 1

Find the values of x, y and z.
Vertical angles practice problem. Find missing angles given one angle.

First, identify the two sets of vertical angles.  Angles x and z are across from each other, so they are a pair of vertical angles.  Angle y and the one marked as 76 degrees are the other set of vertical angles.  Here's a new color coded diagram:
Color coded diagram with vertical angles. Shows two sets of congruent vertical angles.

We also know that vertical angles are always congruent - they always have the same angle measure.  This means that y must be 76 degrees.
Vertical angles are congruent diagram.

Now how do we find x and z?  There's more than one way to find them.  The fastest way is to use the fact that angle x and the angle marked as 76 degrees form a linear pair - they are right next to each other and form a straight line.  This means that they must add up to 180 degrees.  To find angle x, all we need to do is subtract 76 from 180.  This means that x must be 180 - 76 = 104 degrees.
Diagram shows congruent vertical angles and indicates linear pair must add to 180 degrees.

​Now that we know x is 104 degrees, we know that z must also be 104 degrees because x and z are vertical angles.  Vertical angles are always the same so if you find one. you automatically know the other one. No work involved!
Vertical angles are congruent diagram.

Now we know all the angles in the diagram.  Anytime you have two intersecting lines, it will form two sets of congruent angles. As long as you know one of the angles, you can find all the others.
Diagram of intersecting lines with two sets of congruent vertical angles.
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