**Introduction to Triangle Word Problems**

[*The universe*] *cannot be read until we have learnt the language and become familiar with the characters in which it is written*.

*It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word*.

—GALILEO GALILEI (1564–1642)

This lesson provides a review of the classification of triangles by their sides and angles. Triangle word problems will be explained, and you will practice solving them.

**Classifying Triangles by their Angles**

Triangles can be classified, or named, based on their interior angles. The sum of the interior angles of any triangle is always 180°.

**Acute Triangles**

An **acute triangle** is a triangle where each angle measures less than 90°. The following triangle is an acute triangle.

ExampleThe measures of two angles of an acute triangle are 55 and 60, respectively. What is the measure of the third angle?

*Read and understand the question*. This question is looking for the measure of the third angle of a triangle when the measures of two angles are given.

*Make a plan*. The triangle is acute, so the result will be an angle with measure less than 90°. Add the measures of the two known angles and subtract the sum from 180°.

*Carry out the plan*. The sum of the two known angles is 55 + 60 = 115: 180 – 115 = 65. The measure of the third angle is 65°.

*Check your answer*. To check this solution, add the three angles and make sure that the sum is 180°: 55 + 60 + 65 = 180. This solution is checking.

**Obtuse Triangles**

An **obtuse triangle** is a triangle where one angle measures more than 90° but less than 180°.

- The following triangle is an example of an obtuse triangle.

ExampleThe measures of two angles of an obtuse triangle are 110 and 30, respectively. What is the measure of the third angle?

*Read and understand the question*. This question is looking for the measure of the third angle of an obtuse triangle when two angles are given.

*Make a plan*. Add the two known angles and subtract this sum from 180°.

*Carry out the plan*. First, add the two known angles: 110 + 30 = 140. Then, subtract this amount from the total of 180°: 180 – 140 = 40. The third angle is 40°.

*Check your answer*. To check this solution, add the three angles to make sure that the sum is exactly 180°: 110 + 30 + 40 = 180. This answer is checking.

**Right Triangles**

A **right triangle** is a triangle where one angle measures 90°. The sides of the triangle that form the right angle are called the legs, and the side opposite the right angle is called the **hypotenuse**. In a right triangle, the two angles other than the right angle will each be acute.

- The following triangle is a right triangle.

ExampleThe measure of one angle of a right angle is 30°. What is the measure of the other angles?

*Read and understand the question*. This question is asking for the measure of the third angle of a right triangle when one of the other acute angles is known.

*Make a plan*. Add the two known angles and subtract this sum from 180°.

*Carry out the plan*. Because the triangle is a right triangle, one of the other angles is 90°. Add the two known angles: 30 + 90 = 120. Finally, subtract this amount from the total of 180°: 180 – 120 = 60. The third angle is 60°.

*Check your answer*. To check this solution, add the three angles to make sure that the sum is exactly 180°: 30 + 90 + 60 = 180°. This solution is checking.

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