Triangle angle sum

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.

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

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

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

11703 x

Find the measure of each angle indicated.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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6433 x

Solve for x.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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5356 x

Find the measure of angle A.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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Geometry

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Parallel Lines and the Coordinate Plane
Properties of Triangles

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