When two lines cross, they form 4 angles. The angles that are across from each other are called vertical angles. Intersecting lines always form two sets of vertical angles:
Using Vertical Angles
Vertical angles are always congruent - they always have the same angle measure. If one angle is 40 degrees, the vertical angle across from it will also be 40 degrees.
What about the other two angles in the diagram above that aren't labeled? We can use what we know about vertical angles to help us find those angle measures as well. We know that a full circle is always 360 degrees, so if we added all 4 angles together they must add up to 360 degrees. The two 40 degree angles add up to 80 degrees, which means the missing two must add up to 360 - 80 = 280. Since they are also vertical, they must be the same angle measure. So we can simply divide 280 by 2 to figure out each one: 280/2 = 140. This means the missing two angles must be 140 degrees.
There's another way we could have figured out that the missing two angles were 140 degrees. The intersecting lines form sets of linear pairs - angles that are right next to each other that form a straight line. We know that a straight line is always 180 degrees (it's half way around a circle), so we also could use the fact that any two angles right next to each other in the diagram above must also add up to 180 degrees.
You can use what you know about vertical angles and linear pairs to find missing angles formed by intersecting lines.
Example 1 Find the values of x, y and z.
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:
We also know that vertical angles are always congruent - they always have the same angle measure. This means that y must be 76 degrees.
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.
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!
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.
Example 2
The two angles given are vertical angles, so we know they have the same angle measure. This means we can write an equation by setting the two expressions equal to each other.