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.
We learned earlier in this section the basics of an angle. An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In the image below, [latex]\angle A[/latex] is the angle with vertex at point [latex]A[/latex]. We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees.
The measure of [latex]\angle A[/latex] is written [latex]m\angle A[/latex]. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ[/latex], we would write the measure as [latex]m\angle A=\text{27}^ \circ[/latex]. The measures of two angles can be added together. There are two special cases that occur when we add two angle measures: supplementary and complementary angles. 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.
If the sum of the measures of two angles is [latex]\text{90}^ \circ[/latex], then the angles are complementary angles. In the images below, each pair of angles is complementary, because their measures add to [latex]\text{90}^ \circ[/latex]. Each angle is the complement of the other.
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 ones following, you will be introduced to some common geometry formulas. We will adapt our Problem-Solving Strategy for Geometry Applications to solve the problems you will encounter.
Problem-Solving Strategy for Geometry Applications
Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for and choose a variable to represent it.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.
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].
Find its supplement
Find its complement
1.
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
The supplement of a [latex]40°[/latex]angle.
Step 3. Name. Choose a variable to represent it.
Let [latex]s=[/latex]the measure of the supplement.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
The measures of the angle are [latex]75°[/latex]and [latex]105°[/latex].
You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property (see image below) and fence off the area. However, the corner of your property does not form the traditional right angle. You learned from the city that the streets cross at an angle equal to [latex]150^\circ[/latex]. What is the measure of the angle that will border your garden?
As the angle between Linton Street and First Avenue is [latex]150^\circ[/latex], the supplementary angle is [latex]30^\circ[/latex]. Therefore, the garden will form a [latex]30^\circ[/latex] angle at the corner of your property.
