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

Pooja Gupta

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

Learning Objectives

By the end of this section, you will be able to:

Use the properties of angles
Use the properties of triangles
Use the Pythagorean Theorem

Use the Properties of Angles

Are you familiar with the phrase ‘do a $180\text{'?}$ 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 $180$ degrees. See (Figure 1).

Figure 1

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 2), $\angle A$ is the angle with vertex at point $A$ . The measure of $\angle A$ is written $m\angle A$ .

$\angle A$ is the angle with vertex at $\text{point}\phantom{\rule{0.2em}{0ex}}A$ .

The image is an angle made up of two rays. The angle is labeled with letter A. — Figure 2

We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation $m$ to for the measure of an angle. So if $\angle A$ is 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 3), each pair of angles is supplementary because their measures add to $\text{180}$ °. Each angle is the supplement of the other.

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

Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees. — Figure 3

If the sum of the measures of two angles is $\text{90}$ °, then the angles are complementary angles. In (Figure 4), each pair of angles is complementary, because their measures add to $\text{90}$ °. Each angle is the complement of the other.

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

Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees. — Figure 4

Supplementary and Complementary Angles

If the sum of the measures of two angles is $\text{180}$ °, then the angles are supplementary.

If $\angle A$ and $\angle B$ are supplementary, then $m\angle A+m\angle B=\text{180}$ °.

If the sum of the measures of two angles is $\text{90}$ °, then the angles are complementary.

If $\angle A$ and $\angle B$ are complementary, then $m\angle A+m\angle B=\text{90}$ °.

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.

HOW TO: Use a 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.

EXAMPLE 1

An angle measures $\text{40}$ °. Find a) its supplement, and b) its complement.

Solution

a)
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 $40^{\circ}$ angle.
Step 3. Name. Choose a variable to represent it.	let $s =$ the measure of the supplement
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.	$\begin{aligned} & m \angle A+m \angle B=180^{\circ} \\ & s+40^{\circ}=180^{\circ} \end{aligned}$
Step 5. Solve the equation.	$s=140$
Step 6. Check:	$\begin{aligned} & 140^{\circ}+40^{\circ} \stackrel{?}{=} 180^{\circ} \\ & 180^{\circ}= 180^{\circ} \checkmark \end{aligned}$
Step 7. Answer the question.	The supplement of the $40^{\circ}$ angle is $140^{\circ}$

b)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the complement of a $40^{\circ}$ angle.
Step 3. Name. Choose a variable to represent it.	let $c=$ the measure of the complement
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.	$\begin{aligned} &m \angle A+m \angle B=90^\circ \\ & c+40^{\circ}=90^{\circ} \end{aligned}$
Step 5. Solve the equation.	$c=50^\circ$
Step 6. Check:	$\begin{aligned} & 50^{\circ}+40^{\circ} \stackrel{?}{=} 90^{\circ} \\ & 90^{\circ}= 90^{\circ} \checkmark \end{aligned}$
Step 7. Answer the question.	The complement of the $40^\circ$ angle is $50^\circ$ .

TRY IT 1.1

An angle measures $\text{25}$ °. Find its: a) supplement b) complement.

Answer

155°
65°

TRY IT 1.2

An angle measures $\text{77}$ °. Find its: a) supplement b) complement.

Answer

103°
13°

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

EXAMPLE 2

Two angles are supplementary. The larger angle is $\text{30}$ ° more than the smaller angle. Find the measure of both angles.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the measures of both angles
Step 3. Name. Choose a variable to represent it. The larger angle is 30° more than the smaller angle.	let $a =$ measure of smaller angle $a+3 =$ measure of larger angle
Step 4. Translate. Write the appropriate formula and substitute.	$m \angle A+m \angle B=180^\circ$
Step 5. Solve the equation.	$\begin{aligned} (a+30)+a&=180 \\ 2a+30&=180 \\ 2a&=150 \\ {\text { measure of smaller angle } a&=75 \\ \text{measure of larger angle}&= a+30 \\ 75+30 &= 105 \\ \end{aligned}$
Step 6. Check:	$\begin{aligned} & m \angle A + m \angle B = 180^\circ \\& 75^{\circ}+105^{\circ} \stackrel{?}{=} 180^{\circ} \\ & 180^{\circ}= 180^{\circ} \checkmark \end{aligned}$
Step 7. Answer the question.	The measures of angles are $75^\circ$ and $105^\circ$ .

TRY IT 2.1

Two angles are supplementary. The larger angle is $\text{100}$ ° more than the smaller angle. Find the measures of both angles.

Answer

40°, 140°

TRY IT 2.2

Two angles are complementary. The larger angle is $\text{40}$ ° more than the smaller angle. Find the measures of both angles.

Answer

25°, 65°

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

$\Delta ABC$ has vertices $A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C$ and sides $a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c\text{.}$

The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c. — Figure 5

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

$m\angle A+m\angle B+m\angle C=\text{180}$ °

Sum of the Measures of the Angles of a Triangle

For any $\Delta ABC$ , the sum of the measures of the angles is $\text{180}$ °.

$m\angle A+m\angle B+m\angle C=\text{180}$ °

EXAMPLE 3

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.	the measure of the third angle in a triangle
Step 3. Name. Choose a variable to represent it.	let $x=$ the measure of the angle
Step 4. Translate. Write the appropriate formula and substitute.	$m \angle A+m \angle B + m \angle C =180^\circ$
Step 5. Solve the equation.	$\begin{aligned} 55+82+x &=180 \\ 137+x&=180\\ x&=43 \end{aligned}$
Step 6. Check:	$\begin{aligned} & m \angle A + m \angle B +m \angle C = 180^\circ \\& 82^{\circ}+55^{\circ}+43^\circ \stackrel{?}{=} 180^{\circ} \\ & 180^{\circ}= 180^{\circ} \checkmark \end{aligned}$
Step 7. Answer the question.	The measure of the third angle is $43^\circ$ .

TRY IT 3.1

The measures of two angles of a triangle are $\text{31}$ ° and $\text{128}$ °. Find the measure of the third angle.

Answer

21°

TRY IT 3.2

A triangle has angles of $\text{49}$ ° and $\text{75}$ °. Find the measure of the third angle.

Answer

56°

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 6).

A right triangle is shown. The right angle is marked with a box and labeled 90 degrees. — Figure 6

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 4

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.	the measure of an angle
Step 3. Name. Choose a variable to represent it.	Let $x=$ the measure of the angle
Step 4. Translate. Write the appropriate formula and substitute.	$\begin{aligned} m \angle A+m \angle B + m \angle C&=180^\circ \\ 90+28+x=180^\circ \end{aligned}$
Step 5. Solve the equation.	$\begin{aligned} 118+x&=180 \\ x=62^\circ \end{aligned}$
Step 6. Check:	$\begin{aligned} & m \angle A + m \angle B +m \angle C = 180^\circ \\& 90^{\circ}+28^{\circ}+62^\circ \stackrel{?}{=} 180^{\circ} \\ & 180^{\circ}= 180^{\circ} \checkmark \end{aligned}$
Step 7. Answer the question.	The measure of the third angle is $62^\circ.$

TRY IT 4.1

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

Answer

34°

TRY IT 4.2

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

Answer

45°

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 5

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.	Let $a=1^{st}$ angle $a+20=2^{nd}$ angle $90=3^{rd}$ angle (the right angle)
Step 4. Translate. Write the appropriate formula and substitute into the formula.	$\begin{aligned} m \angle A+m \angle B + m \angle C&=180^\circ \\ a + (a+20)+90&=180 \end{aligned}$
Step 5. Solve the equation.	$\begin{aligned} 2a+110 &=180^\circ \\ 2a &=70 \\ a&=35^\circ {\text{first angle}} \\ {\text{second angle}} &= a+20\\ &={\color{red}{35}}+20 \\ &=55^\circ \\ {\text{third angle}} &= 90^\circ \end{aligned}$
Step 6. Check:	$\begin{aligned} & m \angle A + m \angle B +m \angle C = 180^\circ \\& 35^{\circ}+55^{\circ}+90^\circ \stackrel{?}{=} 180^{\circ} \\ & 180^{\circ}= 180^{\circ} \checkmark \end{aligned}$
Step 7. Answer the question.	The three angles measure $35^\circ, 55^\circ \text{and} 90^\circ$ .

TRY IT 5.1

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.

Answer

20°, 70°, 90°

TRY IT 5.2

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.

Answer

30°, 60°, 90°

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 are have the same measures.

The two triangles in (Figure 7) are similar. Each side of $\Delta ABC$ is four times the length of the corresponding side of $\Delta XYZ$ and their corresponding angles have equal measures.

$\Delta ABC$ and $\Delta XYZ$ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3. — Figure 7

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.

Rendered by QuickLaTeX.com

$\begin{aligned} & m \angle A=m \angle X \\ & m \angle B=m \angle Y \\ & m \angle C=m \angle Z \\ & \frac{a}{X}=\frac{b}{y}=\frac{c}{Z} \end{aligned}$

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

$\phantom{\rule{2em}{0ex}}\begin{array}{c}\text{the length}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}BC\hfill \\ \text{the length}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}AC\hfill \\ \text{the length}\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}\text{can also be written}\phantom{\rule{0.2em}{0ex}}AB\hfill \end{array}$

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

EXAMPLE 6

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

Rendered by QuickLaTeX.com

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. $\begin{aligned} & \text { To find } a: \quad \text { To find } y: \\ \text { sides of large triangle } {\color{purple}\longrightarrow} &\frac{A B}{X Y}=\frac{B C}{Y Z} \quad \frac{A B}{X Y}=\frac{A C}{X Z} \\ \text { sides of small triangle } {\color{purple}\longrightarrow} & \quad \frac{4}{3}=\frac{a}{4.5} \quad\quad \frac{4}{3}=\frac{3.2}{y} \\ & \end{aligned}$
Step 5. Solve the equation.	$\begin{array}{rlrl} 3 a & =4(4.5) & 4 y & =3(3.2) \\ 3 a & =18 & 4 y & =9.6 \\ a & =6 & y & =2.4 \end{array}$
Step 6. Check.	$\begin{array}{rlrl} \frac{4}{3} & \stackrel{?}{=} \frac{{\color{red}6}}{4.5} & \frac{4}{3} \stackrel{?}{=} \frac{3.2}{{\color{red}2.4}} \\ 4(4.5) & \stackrel{?}{=} 6(3) & 4(2.4) & \stackrel{?}{=} 3.2(3) \\ 18 & =18 \checkmark & 9.6 & =9.6 \checkmark \end{array}$
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 6.1

$\Delta ABC$ is similar to $\Delta XYZ$ . Find $a$ .

Rendered by QuickLaTeX.com

Answer

8

TRY IT 6.2

$\Delta ABC$ is similar to $\Delta XYZ$ . Find $y$ .

Rendered by QuickLaTeX.com

Answer

22.5

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 8).

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.

Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.” — Figure 8

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 $\Delta ABC$ ,

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

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

Rendered by QuickLaTeX.com

To solve problems that use the Pythagorean Theorem, we will need to find square roots. We defined the notation $\sqrt{m}$ in this way:

$\text{If}\phantom{\rule{0.2em}{0ex}}m={n}^{2},\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\sqrt{m}=n\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}n\ge 0$

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 7

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

Rendered by QuickLaTeX.com

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.	$\begin{aligned} & a^2+b^2=c^2 \\ & 3^2+4^2=c^2 \end{aligned}$
Step 5. Solve the equation.	$\begin{aligned} 9+16 & =\mathrm{c}^2 \\ 25 & =\mathrm{c}^2 \\ \sqrt{25} & =\mathrm{c}^2 \\ 5 & =\mathrm{c} \end{aligned}$
Step 6. Check:	$\begin{aligned} 3^2+4^2 & ={\color{red}5}^2 \\ 9+16 & \stackrel{?}{=} 25 \\ 25 & =25 \checkmark \end{aligned}$
Step 7. Answer the question.	The length of the hypotenuse is 5.

TRY IT 7.1

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

Rendered by QuickLaTeX.com

Answer

10

TRY IT 7.2

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

Rendered by QuickLaTeX.com

Answer

17

EXAMPLE 8

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

Rendered by QuickLaTeX.com

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.	$\begin{aligned} & a^2+b^2=c^2 \\ & 5^2+b^2=13^2 \end{aligned}$
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify.	$\begin{aligned} 25+b^2 & =169 \\ b^2 & =144 \\ b^2 & =\sqrt{144} \\ b & =12 \end{aligned}$
Step 6. Check:	$\begin{aligned} 5^2+{{\color{red}12}^2} & \stackrel{?}{=} 13^2 \\ 25+144 & \stackrel{?}{=} 169 \\ 169 & =169 \end{aligned}$
Step 7. Answer the question.	The length of the leg is 12.

TRY IT 8.1

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

Rendered by QuickLaTeX.com

Answer

8

TRY IT 8.2

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

Rendered by QuickLaTeX.com

Answer

12

EXAMPLE 9

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.

A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.

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.	$\begin{aligned} & a^2+b^2=c^2 \\ & x^2+x^2=10^2 \end{aligned}$
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.	$\begin{aligned} 2 x^2 & =100 \\ x^2 & =50 \\ x & =\sqrt{50} \\ b & \approx 7.1 \end{aligned}$
Step 6. Check:	$\begin{aligned} a^2+b^2 & =c^2 \\ (7.1)^2+(7.1)^2 & \stackrel{?}{\approx} 10^2 \checkmark \end{aligned}$
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1″ from the corner.

TRY IT 9.1

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?

A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.

Answer

12 feet

TRY IT 9.2

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?

A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.

Answer

8 feet

Key Concepts

Supplementary and Complementary Angles
- If the sum of the measures of two angles is 180°, then the angles are supplementary.
- If $\angle A$ and $\angle B$ are supplementary, then $m\angle A+m\angle B=180$ .
- If the sum of the measures of two angles is 90°, then the angles are complementary.
- If $\angle A$ and $\angle B$ are complementary, then $m\angle A+m\angle B=90$ .
Solve Geometry Applications
1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for and choose a variable to represent it.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Sum of the Measures of the Angles of a Triangle
- For any $\Delta ABC$ , the sum of the measures is 180°
- $m\angle A+m\angle B+m\angle C=180$
Right Triangle
- A right triangle is a triangle that has one 90° angle, which is often marked with a ⦜ symbol.
Properties of Similar Triangles
- If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

Glossary

angle: An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.

complementary angles: If the sum of the measures of two angles is $\text{90}$ °, then they are called complementary angles.

hypotenuse: The side of the triangle opposite the 90° angle is called the hypotenuse.

legs of a right triangle: The sides of a right triangle adjacent to the right angle are called the legs.

right triangle: A right triangle is a triangle that has one $\text{90}$ ° angle.

similar figures: In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.

supplementary angles: If the sum of the measures of two angles is $\text{180}$ °, then they are called supplementary angles.

triangle: A triangle is a geometric figure with three sides and three angles.

vertex of an angle: When two rays meet to form an angle, the common endpoint is called the vertex of the angle.

Practice Makes Perfect

Use the Properties of Angles

In the following exercises, find a) the supplement and b) the complement of the given angle.

1. $\text{53}$ °	2. $\text{16}$ °
3. $\text{29}$ °	4. $\text{72}$ °

In the following exercises, use the properties of angles to solve.

5. Find the supplement of a $\text{135}$ ° angle.	6. Find the complement of a $\text{38}$ ° angle.
7. Find the complement of a $27.5$ ° angle.	8. Find the supplement of a $109.5$ ° angle.
9. Two angles are supplementary. The larger angle is $\text{56}$ ° more than the smaller angle. Find the measures of both angles.	10. Two angles are supplementary. The smaller angle is $\text{36}$ ° less than the larger angle. Find the measures of both angles.
11. Two angles are complementary. The smaller angle is $\text{34}$ ° less than the larger angle. Find the measures of both angles.	12. Two angles are complementary. The larger angle is $\text{52}$ ° more than the smaller angle. Find the measures of both angles.

Use the Properties of Triangles

In the following exercises, solve using properties of triangles.

13. The measures of two angles of a triangle are $\text{26}$ ° and $\text{98}$ °. Find the measure of the third angle.	14. The measures of two angles of a triangle are $\text{61}$ ° and $\text{84}$ °. Find the measure of the third angle.
15. The measures of two angles of a triangle are $\text{105}$ ° and $\text{31}$ °. Find the measure of the third angle.	16. The measures of two angles of a triangle are $\text{47}$ ° and $\text{72}$ °. Find the measure of the third angle.
17. One angle of a right triangle measures $\text{33}$ °. What is the measure of the other angle?	18. One angle of a right triangle measures $\text{51}$ °. What is the measure of the other angle?
19. One angle of a right triangle measures $22.5$ °. What is the measure of the other angle?	20. One angle of a right triangle measures $36.5$ °. What is the measure of the other angle?
21. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.	22. The measure of the smallest angle of a right triangle is $\text{20}$ ° less than the measure of the other small angle. Find the measures of all three angles.
23. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.	24. The angles in a triangle are such that the measure of one angle is $\text{20}$ ° more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Find the Length of the Missing Side

In the following exercises, $\Delta ABC$ is similar to $\Delta XYZ$ . Find the length of the indicated side.

25. side $b$

26. side $x$

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 $270$ miles.

A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.

27. Find the distance from Los Angeles to San Francisco.

28. Find the distance from San Francisco to Las Vegas.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

29.	30.
31.	32.

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.

33.	34.
35.	36.
37.	38.
39.	40.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

41. A $\text{13-foot}$ string of lights will be attached to the top of a $\text{12-foot}$ pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?

A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.

42. Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is $12$ feet high and $16$ feet wide. How long should the banner be to fit the garage door?

A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.

43. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of $10$ feet. What will the length of the path be?

A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.

44. Brian borrowed a $\text{20-foot}$ extension ladder to paint his house. If he sets the base of the ladder $6$ feet from the house, how far up will the top of the ladder reach?

A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.

Answers

1. a) 127° b) 37°	3. a) 151° b) 61°	5. 45°
7. 62.5°	9. 62°, 118°	11. 62°, 28°
13. 56°	15. 44°	17. 57°
19. 67.5°	21. 45°, 45°, 90°	23. 30°, 60°, 90°
25. 12	27. 351 miles	29. 15
31. 25	33. 8	35. 12
37. 10.2	39. 8	41. 5 feet
43. 14.1 feet

Attributions

This chapter has been adapted from “Use Properties of Angles, Triangles, and the Pythagorean Theorem” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

Icon for the Creative Commons Attribution 4.0 International License

Use the Properties of Angles

Use the Properties of Triangles

Right Triangles

Similar Triangles

Use the Pythagorean Theorem

Key Concepts

Glossary

Practice Makes Perfect

Use the Properties of Angles

Use the Properties of Triangles

Find the Length of the Missing Side

Use the Pythagorean Theorem

Find the Length of the Missing Side

Answers

Attributions

License

Share This Book