Interior Angles of a Triangles
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Theorem:

The sum of the measures of the interior angles of any triangle is 180º.

 

In at the left,

  m<M + m<N + m<P = 180º.

 

Remember that this theorem works for ANY type of triangle.  The sum of the angles in ANY type of triangle is 180º.

 

Examples

1. In ABC, m<A = 42º and m<C = 63º.  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 = 75º

 

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 30º

 

3. The vertex angle of an isosceles triangle measures 58º.  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 61º.