Interior Angles of a Triangle
Theorem:
The sum of the measures of the interior angles of any triangle is 180°.
In ∆MNP given below,
m∠M + m∠N + m∠P = 180°.
Remember that this theorem works for ANY type of triangle. The sum of the angles in ANY type of triangle is 180°.
Examples
1. In ∆ABC, m∠A = 42° and m∠C = 63°. What is the measure of ∠B ?
Let x = m∠B.
Add up all three angles and set them equal to 180º.
Solve for x.
x + 42 + 63 = 180
x + 105 = 180
x = 75
So m∠B = 75°
2. The angles of a triangle are in the ratio of 1:2:3. Find the measure of the smallest angle of the triangle.
Let x = smallest angle
2x = second angle
3x = largest angle
Then:
x + 2x + 3x = 180
6x = 180
x = 30
So the smallest angle measures 30°
3. The vertex angle of an isosceles triangle measures 58° Find the measure of a base angle.
The base angles are the 2 congruent angles in an isosceles triangle. So, let x = a base angle.
Then
x + x + 58 = 180
2x + 58 = 180
2x = 122
x = 61
So a base angle measures 61°.