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

Use the Properties of Angles

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.

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 $m\angle A.$

Figure 9.6 $\angle A$ is the angle with vertex at $\text{point}A.$

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.

Figure 9.7 The sum of the measures of supplementary angles is $\text{180°}.$

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.

Figure 9.8 The sum of the measures of complementary angles is $\text{90°}.$

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 Figure 9.9 is called $\text{Δ}ABC,$ read ‘triangle $\text{ABC}$’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

Figure 9.9 $\text{Δ}ABC$ has vertices $A,B,\text{and}C$ and sides $a,b,\text{and}c\text{.}$

The three angles of a triangle are related in a special way. The sum of their measures is $\text{180°}.$

Example 9.18

The measures of two angles of a triangle are $\text{55°}$ and $\text{82°}.$ Find the measure of the third angle.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one $\text{90°}$ angle, which is often marked with the symbol shown in Figure 9.10.

Figure 9.10

If we know that a triangle is a right triangle, we know that one angle measures $\text{90°}$ so we only need the measure of one of the other angles in order to determine the measure of the third angle.

Example 9.19

One angle of a right triangle measures $\text{28°}.$ What is the measure of the third angle?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.37

One angle of a right triangle measures $\text{56°}.$ What is the measure of the other angle?

Try It 9.38

One angle of a right triangle measures $\text{45°}.$ What is the measure of the other 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.

Example 9.20

The measure of one angle of a right triangle is $\text{20°}$ more than the measure of the smallest angle. Find the measures of all three angles.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the measures of all three angles
Step 3. Name. Choose a variable to represent it. Now draw the figure and label it with the given information.
Step 4. Translate. Write the appropriate formula and substitute into the formula.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Try It 9.39

The measure of one angle of a right triangle is $\text{50°}$ more than the measure of the smallest angle. Find the measures of all three angles.

Try It 9.40

The measure of one angle of a right triangle is $\text{30°}$ more than the measure of the smallest angle. Find the measures of all three angles.

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 Figure 9.11 are similar. Each side of $\text{Δ}ABC$ is four times the length of the corresponding side of $\text{Δ}XYZ$ and their corresponding angles have equal measures.

Figure 9.11 $\text{Δ}ABC$ and $\text{Δ}XYZ$ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

Properties of Similar Triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $\text{Δ}ABC\text{:}$

$\begin{array}{c}\text{the length}a\text{can also be written}BC\hfill \\ \text{the length}b\text{can also be written}AC\hfill \\ \text{the length}c\text{can also be written}AB\hfill \end{array}$

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

Example 9.21

$\text{Δ}ABC$ and $\text{Δ}XYZ$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	The figure is provided.
Step 2. Identify what you are looking for.	The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it.	Let a = length of the third side of $\Delta ABC$ y = length of the third side $\Delta XYZ$
Step 4. Translate.
The triangles are similar, so the corresponding sides are in the same ratio. So $$\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}$$ Since the side $AB=4$ corresponds to the side $XY=3$, we will use the ratio $\frac{\mathrm{AB}}{\mathrm{XY}}=\frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side of $\Delta ABC$ is 6 and the third side of $\Delta XYZ$ is 2.4.

Try It 9.41

$\text{Δ}ABC$ is similar to $\text{Δ}XYZ.$ Find $a.$

Try It 9.42

$\text{Δ}ABC$ is similar to $\text{Δ}XYZ.$ Find $y.$

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 $500$ BCE.

Remember that a right triangle has a $\text{90°}$ angle, which we usually mark with a small square in the corner. The side of the triangle opposite the $\text{90°}$ 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 $\text{90°}$ 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.

The Pythagorean Theorem

In any right triangle $\text{Δ}ABC,$

$$a^2+b^2=c^2$$

where $c$ is the length of the hypotenuse $a$ and $b$ are the lengths of the legs.

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 $\sqrt{m}$ and defined it in this way:

For example, we found that $\sqrt{25}$ is $5$ because $5^2=25.$

We will use this definition of square roots to solve for the length of a side in a right triangle.

Example 9.22

Use the Pythagorean Theorem to find the length of the hypotenuse.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it.	Let $c=\text{the length of the hypotenuse}$
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the hypotenuse is 5.

Try It 9.43

Use the Pythagorean Theorem to find the length of the hypotenuse.

Try It 9.44

Use the Pythagorean Theorem to find the length of the hypotenuse.

Example 9.23

Use the Pythagorean Theorem to find the length of the longer leg.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	The length of the leg of the triangle
Step 3. Name. Choose a variable to represent it.	Let $b=\text{the leg of the triangle}$ Label side b
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify.
Step 6. Check:
Step 7. Answer the question.	The length of the leg is 12.

Try It 9.45

Use the Pythagorean Theorem to find the length of the leg.

Try It 9.46

Use the Pythagorean Theorem to find the length of the leg.

Example 9.24

Kelvin is building a gazebo and wants to brace each corner by placing a $\text{10-inch}$ wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = the distance from the corner
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.
Step 6. Check: Yes.
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Try It 9.47

John puts the base of a $\text{13-ft}$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

Try It 9.48

Randy wants to attach a $\text{17-ft}$ string of lights to the top of the $\text{15-ft}$ mast of his sailboat. How far from the base of the mast should he attach the end of the light string?