### Measuring Angles

We measure a line segment’s *length* with a ruler. Angles are measured with something called a **protractor**. A protractor is a measuring device that measures how “open” an angle is. Angles are measured in degrees, and labeled with a \begin{align*}^\circ\end{align*} symbol.

Notice that there are two sets of measurements, one opening clockwise and one opening counter-clockwise, from \begin{align*}0^\circ\end{align*} to \begin{align*}180^\circ\end{align*}. When measuring angles, always line up one side with \begin{align*}0^\circ\end{align*}, and see where the other side hits the protractor. The vertex lines up in the middle of the bottom line, where all the degree lines meet.

For every angle there is a number between \begin{align*}0^\circ\end{align*} and \begin{align*}180^\circ\end{align*} that is the measure of the angle in degrees. The angle's measure is then the absolute value of the difference of the numbers shown on the protractor where the sides of the angle intersect the protractor. In other words, you do not have to start measuring an angle at \begin{align*}0^\circ\end{align*}, as long as you subtract one measurement from the other.

The **Angle Addition Postulate** states that if \begin{align*}B\end{align*} is on the interior of \begin{align*}\angle ADC\end{align*}, then \begin{align*}m \angle ADC = m \angle ADB + m \angle BDC\end{align*}. See the picture below.

#### Drawing a \begin{align*}50^\circ\end{align*} Angle with a Protractor

- Start by drawing a horizontal line across the page, about 2 in long.
- Place an endpoint at the left side of your line.
- Place the protractor on this point. Make sure to put the center point on the bottom line of the protractor on the vertex. Mark \begin{align*}50^\circ\end{align*} on the appropriate scale.
- Remove the protractor and connect the vertex and the \begin{align*}50^\circ\end{align*} mark.

#### Copying an Angle with a Compass and Straightedge

- We are going to copy the angle created in the previous investigation, a \begin{align*}50^\circ\end{align*} angle. First, draw a straight line, about 2 inches long, and place an endpoint at one end.
- With the point (non-pencil side) of the compass on the vertex, draw an arc that passes through both sides of the angle. Repeat this arc with the line we drew in #1.
- Move the point of the compass to the horizontal side of the angle we are copying. Place the point where the arc intersects this side. Open (or close) the “mouth” of the compass so you can draw an arc that intersects the other side of the arc drawn in #2. Repeat this on the line we drew in #1.
- Draw a line from the new vertex to the arc intersections.

#### Measuring Angles

Measure the three angles using a protractor.

It might be easier to measure these three angles if you separate them. With measurement, we put an \begin{align*}m\end{align*} in front of the \begin{align*}\angle\end{align*} sign to indicate measure. So, \begin{align*}m \angle XUY = 84^\circ, \ m \angle YUZ = 42^\circ\end{align*} and \begin{align*}m \angle XUZ = 126^\circ\end{align*}.

#### Using a Protractor

What is the measure of the angle shown below?

This angle is not lined up with \begin{align*}0^\circ\end{align*}, so use subtraction to find its measure. It does not matter which scale you use.

Using the inner scale, \begin{align*}|140 - 25| = 125^\circ\end{align*}

Using the outer scale, \begin{align*}|165 - 40| = 125^\circ\end{align*}

#### Using the Angle Addition Postulate

What is \begin{align*}m \angle QRT\end{align*} in the diagram below?

Using the Angle Addition Postulate, \begin{align*}m \angle QRT = 15^\circ + 30^\circ = 45^\circ\end{align*}.

#### Drawing an Angle

Draw a \begin{align*}135^\circ\end{align*} angle.

Following the steps from above, your angle should look like this:

### Examples

#### Example 1

Use a protractor to measure \begin{align*}\angle RST\end{align*} below.

The easiest way to measure any angle is to line one side up with \begin{align*}0^\circ\end{align*}. This angle measures \begin{align*}100^\circ\end{align*}.

#### Example 2

What is \begin{align*}m \angle LMN\end{align*} if \begin{align*}m \angle LMO = 85^\circ\end{align*} and \begin{align*}m \angle NMO = 53^\circ\end{align*}?

From the Angle Addition Postulate, \begin{align*} m \angle LMO = m \angle NMO + m \angle LMN\end{align*}. Substituting in what we know, \begin{align*}85^\circ = 53^\circ + m \angle LMN\end{align*}, so \begin{align*}85^\circ - 53^\circ = m \angle LMN\end{align*} or \begin{align*}m \angle LMN = 32^\circ\end{align*}.

#### Example 3

If \begin{align*}m \angle ABD = 100^\circ\end{align*}, find \begin{align*}x\end{align*} and \begin{align*}m \angle ABC\end{align*} and \begin{align*}m \angle CBD\end{align*}?

From the Angle Addition Postulate, \begin{align*}m \angle ABD = m \angle ABC + m \angle CBD\end{align*}. Substitute in what you know and solve the equation.

\begin{align*}100^\circ & = (4x + 2)^\circ + (3x - 7)^\circ\\ 100^\circ & = 7x - 5^\circ\\ 105^\circ & = 7x\\ 15^\circ & = x\end{align*}

So, \begin{align*}m \angle ABC = 4(15^\circ) + 2^\circ = 62^\circ\end{align*} and \begin{align*}m \angle CBD = 3(15^\circ) - 7^\circ = 38^\circ\end{align*}.

### Review

1. What is \begin{align*}m \angle LMN\end{align*} if \begin{align*}m \angle LMO = 85^\circ\end{align*} and \begin{align*}m \angle NMO = 53^\circ\end{align*}?

2. If \begin{align*}m\angle ABD = 100^\circ\end{align*}, find \begin{align*}x\end{align*}.

For questions 3-6, determine if the statement is true or false.

- For an angle \begin{align*}\angle ABC, C\end{align*} is the vertex.
- For an angle \begin{align*}\angle ABC, \overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*} are the sides.
- The \begin{align*}m\end{align*} in front of \begin{align*}m \angle ABC\end{align*} means measure.
- The Angle Addition Postulate says that an angle is equal to the sum of the smaller angles around it.

For 7-12, draw the angle with the given degree, using a protractor and a ruler.

- \begin{align*}55^\circ\end{align*}
- \begin{align*}92^\circ\end{align*}
- \begin{align*}178^\circ\end{align*}
- \begin{align*}5^\circ\end{align*}
- \begin{align*}120^\circ\end{align*}
- \begin{align*}73^\circ\end{align*}

For 13-16, use a protractor to determine the measure of each angle.

Solve for \begin{align*}x\end{align*}.

- \begin{align*}m\angle ADC = 56^\circ\end{align*}
- \begin{align*}m \angle ADC = 130^\circ\end{align*}
- \begin{align*}m \angle ADC = (16x - 55)^\circ\end{align*}
- \begin{align*}m \angle ADC = ( 9x - 80)^\circ\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 1.5.