http://www.ck12.org Chapter 1. Basics of Geometry
Investigation: Vertical Angle Relationships
- Draw two intersecting lines on your paper. Label the four angles created^61 ,^62 ,^63 ,and^6 4. See the picture
above. - Take your protractor and findm^6 1.
- What is the angle relationship between^6 1 and^6 2? Findm^6 2.
- What is the angle relationship between^6 1 and^6 4? Findm^6 4.
- What is the angle relationship between^6 2 and^6 3? Findm^6 3.
- Are any angles congruent? If so, write down the congruence statement.
From this investigation, hopefully you found out that^61 ∼=^6 3 and^62 ∼=^6 4. This is our first theorem. That means it
must be proven true in order to use it.
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.