第9章数学モデルと幾何

So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems.

Are you familiar with the phrase ‘do a $180\text{’?}$ It means to 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 $180$ degrees. See Figure 9.5.

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 Figure 9.6, $\text{∠}A$ is the angle with vertex at point $A.$ The measure of $\text{∠}A$ is written $m\angle A.$

We measure angles in degrees, and use the symbol $°$ to represent degrees. We use the abbreviation $m$ for the measure of an angle. So if $\text{∠}A$ is $\text{27°},$ we would write $m\angle A=27.$

If the sum of the measures of two angles is $\text{180°},$ then they are called supplementary angles. In Figure 9.7, each pair of angles is supplementary because their measures add to $\text{180°}.$ Each angle is the supplement of the other.

If the sum of the measures of two angles is $\text{90°},$ then the angles are complementary angles. In Figure 9.8, each pair of angles is complementary, because their measures add to $\text{90°}.$ Each angle is the complement of the other.

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.

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

Did you notice that the words complementary and supplementary are in alphabetical order just like $90$ and $180$ are in numerical order?