http://www.ck12.org Chapter 1. Basics of Geometry
Vertical Angles Theorem:If two angles are vertical angles, then they are congruent.
We can prove the Vertical Angles Theorem using the same process we used above. However, let’s not use any
specific values for the angles.
From the picture above:(^6) 1 and (^6) 2 are a linear pair m (^61) +m (^62) = 180 ◦
(^6) 2 and (^6) 3 are a linear pair m (^62) +m (^63) = 180 ◦
(^6) 3 and (^6) 4 are a linear pair m (^63) +m (^64) = 180 ◦
All of the equations= 180 ◦,so set the m^61 +m^62 =m^62 +m^63
first and second equation equal to AND
each other and the second and third. m^62 +m^63 =m^63 +m^64
Cancel out the like terms m^61 =m^63 ,m^62 =m^64
Recall that anytime the measures of two angles are equal, the angles are also congruent.
Example A
Findm^6 1 andm^6 2.
(^6) 1 is vertical angles with 18◦, som (^61) = 18 ◦. (^6) 2 is a linear pair with (^6) 1 or 18◦, so 18◦+m (^62) = 180 ◦.m (^62) =
180 ◦− 18 ◦= 162 ◦.
Example B
Name one pair of vertical angles in the diagram below.
One example is^6 INJand^6 MNL.
Example C
If^6 ABCand^6 DBFare vertical angles andm^6 ABC= ( 4 x+ 10 )◦andm^6 DBF= ( 5 x+ 2 )◦, what is the measure of
each angle?
Vertical angles are congruent, so set the angles equal to each other and solve forx. Then go back to find the measure
of each angle.
4 x+ 10 = 5 x+ 2
x= 8
So,m^6 ABC=m^6 DBF= ( 4 ( 8 )+ 10 )◦= 42 ◦
Watch this video for help with the Examples above.