Vertical angles as congruent angles

 
 
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Vertical angles are congruent angles

In this lesson we’ll look at how to use vertical angles to solve problems.

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Vertical angles

Vertical angles are angles in opposite corners of intersecting lines, like these:

 
vertical angles in intersecting lines
 

So vertical angles always share the same vertex, or corner point of the angle. They’re a special angle pair because their measures are always equal to one another. Therefore, from the diagram we can say

???a{}^\circ =b{}^\circ???

or

???\angle a\cong \angle b???

 
 

How to use vertical (and congruent) angles to solve geometry problems


 
 
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Vertical angles that include an unknown value

Example

Find the value of ???x???.

 
Finding the value of an unknown in vertical angles
 


???58{}^\circ??? and ???(10x+18){}^\circ??? are vertical angles, and are therefore congruent, so we can set them equal to one another and solve for the variable.

???(10x+18){}^\circ =58{}^\circ???

???10x{}^\circ =40{}^\circ???

???x=4{}^\circ???


Let’s try another one.


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Vertical angles are angles in opposite corners of intersecting lines.

Example

Solve for the variable.

 
vertical angles in triangles
 


Use the fact that vertical angles are congruent to complete the triangle on the right. The angle measuring ???(4x+8)^\circ??? has a vertical angle inside the triangle on the right.

 
identifying vertical angles in triangles
 

The angles in a triangle sum to ???180{}^\circ???, so we can set up an equation to solve for the variable.

???(4x+8){}^\circ +(3x+12){}^\circ +90{}^\circ =180{}^\circ???

???7x{}^\circ +110{}^\circ =180{}^\circ???

???7x{}^\circ =70{}^\circ???

???x=10{}^\circ???

 
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