Introduction: Angles that can be parts of a triangle can be classified in three different ways. We show and define these types of angles. We assume the facts that the three angles in a triangle as well as any angles that form a straight line have a sum of 180º. Also a “square corner” has a measure of 90º.

The Lesson:

Angles are named and measured according to certain standards. An angle is formed when two lines or rays intersect. Examples and comments about the name and measure of the angle are shown below.

1. Two rays intersecting at point A

In example 1, we can name this angle A or . There is only one angle. Its measurement can be determined by using a protractor. But clearly the measure is less than 90º since this is not as wide an angle as a right angle or square corner.

2. Three lines intersecting at point P

In example 2, there are several angles formed and they must be named carefully to avoid confusion. We can name by the three letters B, P, and C making sure that the vertex P is the middle letter: We see that . Similarly . Together, angles 1 and 2 form which is clearly larger than a square corner; therefore its measure will be larger than 90º.

The angles in a triangle are all between in measure. We classify angles into three types as follows:

• If an angle A has measure , then is said to be acute.

• If an angle A has measure exactly , then is said to be a right angle.

• If an angle A has measure , then is said to be an obtuse angle.

Let's Practice:

1. Suppose a right triangle ABC with angle A = 90º has one acute angle C of 34º. Describe and measure the third angle B?

The third angle B is also acute and is found by subtracting from as follows:

2. A triangle XYZ has . Describe and measure .

We calculate the measure of as follows:

Therefore is acute.

3. In the diagram below, suppose has a measure of 128º and has a measure of 80º. What is the measure of ?

Therefore

We subtract 80 and get

Note that we say “an angle CPD has a measure of 48” while we say . Whether or not we use the degree symbol depends on the context and usually upon whether the word “measure” is used. This varies depending on different textbooks as well.