In any triangle, there are always three interior angles. These inside angles always add up to 180°. This rule is very helpful in finding missing angles in a triangle.

Example 1:

What is \(\angle {\text{B}}\)?

All three angles have to add to 180°, so we have:

\(\angle {\text{B }} + {\text{ 31 }} + {\text{ 45 }} = {\text{ 18}}0\)

\(\angle {\text{B }} + {\text{ 76 }} = {\text{ 18}}0\)         (combine like terms)

\(\angle {\text{B }} = {\text{ 1}}0{\text{4}}^\circ \)

Example 2:

What is \(\angle {\text{D}}\)?

This is a right triangle, so \(\angle {\text{E }} = {\text{ 9}}0^\circ \).

\(\angle {\text{D }} + {\text{ 9}}0{\text{ }} + {\text{ 29 }} = {\text{ 18}}0\)

\(\angle {\text{D }} + {\text{ 119 }} = {\text{ 18}}0\)

\(\angle {\text{D }} = {\text{ 61}}^\circ \)

Example 3: Sometimes, you’ll need to use this property to solve for a variable.

Solve for x.

We know that all the angles have to equal 180°.

\({\text{65 }} + {\text{ 4}}0{\text{ }} + {\text{ x }} + {\text{ 83 }} = {\text{ 18}}0\)

\({\text{188 }} + {\text{ x }} = {\text{ 18}}0\)

                                                  \({\text{x }} = {\text{ }} - {\text{8}}\)

It’s okay that x is a negative number. The angles in a triangle, however, should not be negative. Let’s plug in our answer to make sure this is the case and to check our result.

\({\text{65 }} + {\text{ 4}}0{\text{ }} + {\text{ }}\left( { - {\text{8 }} + {\text{ 83}}} \right){\text{ }} = {\text{ 18}}0\)

\({\text{65 }} + {\text{ 4}}0{\text{ }} + {\text{ 75 }} = {\text{ 18}}0\)

\({\text{18}}0{\text{ }} = {\text{ 18}}0\) ✓

Example 4: Sometimes, we won’t know any of the angles to start with!

Find all three angles.

We can still use the fact that they have to add to 180°to figure this out.

\({\text{3x }} + {\text{ 28 }} + {\text{ 5x }} + {\text{ 52 }} + {\text{ 2x }}--{\text{ 1}}0{\text{ }} = {\text{ 18}}0\)

\({\text{1}}0{\text{x }} + {\text{ 7}}0{\text{ }} = {\text{ 18}}0\)

\({\text{1}}0{\text{x}} = {\text{11}}0\)

\({\text{x }} = {\text{ 11}}\)

Plug in x = 11 into all the angles to find their measures.

\(\angle {\text{A }} = {\text{ 3x }} + {\text{ 28}}\) ►\({\text{3}}\left( {{\text{11}}} \right){\text{ }} + {\text{ 28}}\) ► \({\text{33 }} + {\text{ 28 }} = {\text{ 61}}^\circ \)

\(\angle {\text{B }} = {\text{ 5x }} + {\text{ 52}}\) ► \({\text{5}}\left( {{\text{11}}} \right){\text{ }} + {\text{ 52}}\) ► \({\text{55 }} + {\text{ 52 }} = {\text{ 1}}0{\text{7}}^\circ \)

\(\angle {\text{C }} = {\text{ 2x }}-{\text{ 1}}0\) ► \({\text{2}}\left( {{\text{11}}} \right){\text{ }}-{\text{ 1}}0\) ► \({\text{22 }}-{\text{ 1}}0{\text{ }} = {\text{ 12}}^\circ \)

Below you can download some free math worksheets and practice.