CK-12 Geometry - Second Edition

4.1. Triangle Sums http://www.ck12.org

Triangle Sum Theorem:The interior angles of a triangle add up to 180◦.

Example 1:What is them^6 T?

Solution:From the Triangle Sum Theorem, we know that the three angles add up to 180◦. Set up an equation to
solve for^6 T.

m^6 M+m^6 A+m^6 T= 180 ◦

82 ◦+ 27 ◦+m^6 T= 180 ◦

109 ◦+m^6 T= 180 ◦

m^6 T= 71 ◦

Investigation 4-1 is one way to show that the angles in a triangle add up to 180◦. However, it is not a two-column
proof. Here we will prove the Triangle Sum Theorem.

Given: 4 ABCwith

←→

AD||BC

Prove:m^61 +m^62 +m^63 = 180 ◦

TABLE4.1:

Statement Reason

1. 4 ABCabove with

←→

AD||BC Given

2.^61 ∼=^64 ,^62 ∼=^65 Alternate Interior Angles Theorem
3.m^61 =m^64 ,m^62 =m^65 ∼=angles have = measures
4.m^64 +m^6 CAD= 180 ◦ Linear Pair Postulate
5.m^63 +m^65 =m^6 CAD Angle Addition Postulate
6.m^64 +m^63 +m^65 = 180 ◦ Substitution PoE
7.m^61 +m^63 +m^62 = 180 ◦ Substitution PoE

Example 2:What is the measure of each angle in an equiangular triangle?