4.1. Triangle Sums http://www.ck12.org
130 ◦+ 110 ◦+x= 360 ◦
x= 360 ◦− 130 ◦− 110 ◦
x= 120 ◦xandpare supplementary and add up to 180◦.
x+p= 180 ◦
120 ◦+p= 180 ◦
p= 60 ◦Exterior Angles Theorem
Remote Interior Angles:The two angles in a triangle that are not adjacent to the indicated exterior angle.
(^6) Aand (^6) Bare the remote interior angles for exterior angle (^6) ACD.
Example 7:Findm^6 A.
Solution:First, findm^6 ACB.m^6 ACB+ 115 ◦= 180 ◦by the Linear Pair Postulate, som^6 ACB= 65 ◦.
m^6 A+ 65 ◦+ 79 ◦= 180 ◦by the Triangle Sum Theorem, som^6 A= 36 ◦.
In Example 7,m^6 A+m^6 Bis 36◦+ 79 ◦= 115 ◦. This is the same as the exterior angle atC, 115◦.
From this example, we can conclude the Exterior Angle Theorem.
Exterior Angle Theorem:The sum of the remote interior angles is equal to the non-adjacent exterior angle.
From the picture above, this means thatm^6 A+m^6 B=m^6 ACD.
Here is the proof of the Exterior Angle Theorem. From the proof, you can see that this theorem is a combination of
the Triangle Sum Theorem and the Linear Pair Postulate.
Given: 4 ABCwith exterior angle^6 ACD
Prove:m^6 A+m^6 B=m^6 ACD