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Appendix B: Geometry

Using properties of angles to solve problems, learning outcomes.

Are you familiar with the phrase ‘do a [latex]180[/latex]?’ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is [latex]180[/latex] degrees. See the image below.

The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees.

[latex]\angle A[/latex] is the angle with vertex at [latex]\text{point }A[/latex].

The image is an angle made up of two rays. The angle is labeled with letter A.

We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] to for the measure of an angle. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ [/latex], we would write [latex]m\angle A=27[/latex].

If the sum of the measures of two angles is [latex]\text{180}^ \circ[/latex], then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to [latex]\text{180}^ \circ [/latex]. Each angle is the supplement of the other.

The sum of the measures of supplementary angles is [latex]\text{180}^ \circ [/latex].

Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.

The sum of the measures of complementary angles is [latex]\text{90}^ \circ[/latex].

Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.

Supplementary and Complementary Angles

If the sum of the measures of two angles is [latex]\text{180}^\circ [/latex], then the angles are supplementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are supplementary, then [latex]m\angle{A}+m\angle{B}=180^\circ[/latex].

If the sum of the measures of two angles is [latex]\text{90}^\circ[/latex], then the angles are complementary .

If angle [latex]A[/latex] and angle [latex]B[/latex] are complementary, then [latex]m\angle{A}+m\angle{B}=90^\circ[/latex].

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

Use a Problem Solving Strategy for Geometry Applications.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

An angle measures [latex]\text{40}^ \circ[/latex].

1. Find its supplement

2. Find its complement

Write the appropriate formula for the situation and substitute in the given information. [latex]m\angle A+m\angle B=90[/latex] Step 5. Solve the equation. [latex]c+40=90[/latex]

[latex]c=50[/latex] Step 6. Check:


In the following video we show more examples of how to find the supplement and complement of an angle.

Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[/latex] and [latex]180[/latex] are in numerical order?

Two angles are supplementary. The larger angle is [latex]\text{30}^ \circ[/latex] more than the smaller angle. Find the measure of both angles.

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