By the end of this section, you will be able to:
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’? 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, ∠A is the angle with vertex at point A. The measure of ∠A is written m∠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 ∠A is 27°, we would write m∠A=27.
If the sum of the measures of two angles is 180°, then they are called supplementary angles. In Figure 9.7, each pair of angles is supplementary because their measures add to 180°. Each angle is the supplement of the other.
If the sum of the measures of two angles is 90°, then the angles are complementary angles. In Figure 9.8, each pair of angles is complementary, because their measures add to 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.
Step 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name what you are looking for and choose a variable to represent it.
Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. 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.
Did you notice that the words complementary and supplementary are in alphabetical order just like 90 and 180 are in numerical order?
What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in Figure 9.9 is called ΔABC, read ‘triangle ABC’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.
Figure 9.9 ΔABC has vertices A,B,andC and sides a,b,andc.
The three angles of a triangle are related in a special way. The sum of their measures is 180°.
If we know that a triangle is a right triangle, we know that one angle measures 90° so we only need the measure of one of the other angles in order to determine the measure of the third angle.
In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.
When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.
The two triangles in Figure 9.11 are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.
Figure 9.11 ΔABC and ΔXYZ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.
The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in ΔABC:
We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.
The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around 500 BCE.
Remember that a right triangle has a 90° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° angle is called the hypotenuse, and the other two sides are called the legs. See Figure 9.12.
Figure 9.12 In a right triangle, the side opposite the 90° angle is called the hypotenuse and each of the other sides is called a leg.
The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.
Use the Properties of Angles
In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?