# Math Models and Geometry

## Use Properties of Angles, Triangles, and the Pythagorean Theorem

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.

### Use the Properties of Angles

Are you familiar with the phrase ‘do a  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  degrees. See .
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 ,  is the angle with vertex at point  The measure of  is written 
We measure angles in degrees, and use the symbol  to represent degrees. We use the abbreviation  for the measure of an angle. So if  is  we would write 
If the sum of the measures of two angles is  then they are called supplementary angles. In , each pair of angles is supplementary because their measures add to  Each angle is the supplement of the other.
If the sum of the measures of two angles is  then the angles are complementary angles. In , each pair of angles is complementary, because their measures add to  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  and  are in numerical order?

### Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in  is called  read ‘triangle ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.
The three angles of a triangle are related in a special way. The sum of their measures is 

### Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one  angle, which is often marked with the symbol shown in .
If we know that a triangle is a right triangle, we know that one angle measures  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.

### Similar Triangles

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 have the same measures.
The two triangles in  are similar. Each side of  is four times the length of the corresponding side of  and their corresponding angles have equal measures.
The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in 

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

### Use the Pythagorean Theorem

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  BCE.
Remember that a right triangle has a  angle, which we usually mark with a small square in the corner. The side of the triangle opposite the  angle is called the hypotenuse, and the other two sides are called the legs. See .
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.
To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation  and defined it in this way:
For example, we found that  is  because 
We will use this definition of square roots to solve for the length of a side in a right triangle.

### Key Concepts

1. Supplementary and Complementary Angles
2. Solve Geometry Applications
3. Sum of the Measures of the Angles of a Triangle
4. Right Triangle

5. Properties of Similar Triangles

### Practice Makes Perfect

Use the Properties of Angles 
In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle.
In the following exercises, use the properties of angles to solve.
Use the Properties of Triangles 
In the following exercises, solve using properties of triangles.
Find the Length of the Missing Side 
In the following exercises,  is similar to  Find the length of the indicated side.
On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is  miles. 



Use the Pythagorean Theorem 
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
Find the Length of the Missing Side 
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.
In the following exercises, solve. Approximate to the nearest tenth, if necessary.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ 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?
