4.1. Triangle Sum Theorem http://www.ck12.org
Investigation: Triangle Tear-Up
Tools Needed: paper, ruler, pencil, colored pencils
- Draw a triangle on a piece of paper. Try to make all three angles different sizes. Color the three interior angles
three different colors and label each one,^61 ,^62 ,and^6 3. - Tear off the three colored angles, so you have three separate angles.
- Attempt to line up the angles so their points all match up. What happens? What measure do the three angles
add up to?
This investigation shows us that the sum of the angles in a triangle is 180◦because the three angles fit together to
form a straight line. Recall that a line is also a straight angle and all straight angles are 180◦.
TheTriangle Sum Theoremstates that the interior angles of a triangle add up to 180◦. The above investigation 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- 4 ABCabove with
←→
AD||BC Given2.^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
There are two theorems that we can prove as a result of the Triangle Sum Theorem and our knowledge of triangles.
Theorem #1:Each angle in an equiangular triangle measures 60◦.
Theorem #2:The acute angles in a right triangle are always complementary.
Example A
What is them^6 T?
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 ◦Example B
Show why Theorem #1 is true.