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

Jared Eusea; Karen Perilloux; Kristina VanDusen; Lauren Johnson; Prakash Ghimire

1.1 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

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 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 1.

Figure 1 A [latex]180^{\circ} [/latex] angle.

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, [latex]\angle A[/latex] is the angle with vertex at point [latex]A[/latex].

The measure of [latex]\angle A[/latex] is written [latex]m\angle A[/latex].

The image is an angle made up of two rays. The angle is labeled with letter A. — **Figure 2**
[latex]\angle A[/latex] is the angle with vertex at point [latex]A[/latex].

We measure angles in degrees, and use the symbol [latex]^{\circ}[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] for the measure of an angle. So if [latex]\angle A[/latex] is [latex]27^{\circ}[/latex], we would write [latex]m\angle A=27[/latex].

If the sum of the measures of two angles is [latex]180^{\circ}[/latex], then they are called supplementary angles. In Figure 3, each pair of angles is supplementary because their measures add to [latex]180^{\circ}[/latex].

Each angle is the supplement of the other.

Part a shows a 120° angle next to a 60° angle. Together, the angles form a straight line and 180°. Part b shows a 45° angle attached to a 135° angle. — **Figure 3**
(a) [latex]120^{\circ}+60^{\circ}=180^{\circ}[/latex] (b) [latex]45^{\circ}+135^{\circ}=180^{\circ}[/latex]

If the sum of the measures of two angles is [latex]90^{\circ}[/latex], then the angles are complementary angles. In Figure 4, each pair of angles is complementary, because their measures add to [latex]90^{\circ}[/latex].

Each angle is the complement of the other.

Part a shows a 50°angle next to a 40°angle. Together, the angles form a right angle. Part b shows a 60° angle attached to a 30° angle. 60° + 30° =90°. — **Figure 4** The sum of the measures of complementary angles is [latex]90^{\circ}[/latex].

Supplementary and Complementary Angles

If the sum of the measures of two angles is [latex]180^{\circ}[/latex], then the angles are supplementary.

If [latex]\angle A[/latex] and [latex]\angle B[/latex] are supplementary, then [latex]m\angle A+m\angle B=180^{\circ}[/latex].

If the sum of the measures of two angles is [latex]90^{\circ}[/latex], then the angles are complementary.

If [latex]\angle A[/latex] and [latex]\angle B[/latex] are complementary, then [latex]m\angle A+m\angle B=90^{\circ}[/latex].

When solving applications involving 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.

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.

Finding the Supplement and Complement of an Angle

An angle measures [latex]40^{\circ}[/latex].

Find (a) its supplement, and (b) its complement.

Show Solution to (a)

Steps	Explanation
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^° 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.	[latex]m\angle A + m\angle B = 180[/latex] [latex]s + 40 = 180[/latex]
Step 5. Solve the equation.	[latex] s = 140[/latex]
Step 6. Check:	[latex]140 + 40 \stackrel{?}{=} 180[/latex] [latex]✔ \qquad 180 = 180[/latex]
Step 7. Answer the question.	The supplement of the 40^° angle is 140°.

Show Solution to (b)

Steps	Explanation
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^° 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.	[latex]m\angle A + m\angle B = 90[/latex] [latex]c + 40 = 90[/latex]
Step 5. Solve the equation.	c = 50
Step 6. Check:	[latex]50 + 40 \stackrel{?}{=} 90[/latex] [latex]✔ \qquad 90 = 90[/latex]
Step 7. Answer the question.	The complement of the 40^° angle is 50°.

Try It

An angle measures [latex]25^{\circ}[/latex].

Find its: (a) supplement (b) complement.

Show Solution

(a) The supplement is [latex]155^{\circ}[/latex].

(b) The complement is [latex]65^{\circ}[/latex].

Try It

An angle measures [latex]77^{\circ}[/latex].

Find its: (a) supplement (b) complement.

Show Solution

(a) The supplement is [latex]103^{\circ}[/latex].

(b) The complement is [latex]13^{\circ}[/latex].

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

Finding Measure of Supplementary Angles

Two angles are supplementary. The larger angle is [latex]30^{\circ}[/latex] more than the smaller angle. Find the measure of both angles.

Show Solution

Steps	Explanation
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 + 30 = measure of larger angle
Step 4. Translate. Write the appropriate formula and substitute.	[latex]m\angle A + m\angle B = 180[/latex]
Step 5. Solve the equation.	[latex](a+30)+a=180[/latex] [latex]2a+30=180[/latex] [latex]2a=150[/latex] [latex]a=75[/latex] measure of smaller angle [latex]a+30[/latex] measure of larger angle [latex]75+30 = 105[/latex]
Step 6. Check:	[latex]m\angle A + m\angle B = 180[/latex] [latex]75 + 105 \stackrel{?}{=} 180[/latex] [latex]✔ \qquad 180 = 180[/latex]
Step 7. Answer the question.	The measures of the angles are 75^° and 105°.

Try It

Two angles are supplementary. The larger angle is [latex]100^{\circ}[/latex] more than the smaller angle. Find the measures of both angles.

Show Solution

The angles have measures [latex]40^{\circ}[/latex] and [latex]140^{\circ}[/latex].

Try It

Two angles are complementary. The larger angle is [latex]40^{\circ}[/latex] more than the smaller angle. Find the measures of both angles.

Show Solution

The angles have measures [latex]25^{\circ}[/latex] and [latex]65^{\circ}[/latex].

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 [latex]\triangle ABC[/latex], read ‘triangle ABC’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c. — **Figure 5**
A [latex]180^{\circ} [/latex] angle.

[latex]\triangle ABC[/latex] has vertices [latex]A, B[/latex], and [latex]C[/latex] and sides [latex]a, b[/latex], and [latex]c[/latex].

The three angles of a triangle are related in a special way. The sum of their measures is [latex]180^{\circ}[/latex].

[latex]m\angle A +m\angle B + m\angle C =180^{\circ}[/latex]

Sum of the Measures of the Angles of a Triangle

For any [latex]\triangle ABC[/latex], the sum of the measures of the angles is [latex]180^{\circ}[/latex].

[latex]m\angle A +m\angle B + m\angle C =180^{\circ}[/latex]

Finding Measure of Third Angle of a Triangle

The measures of two angles of a triangle are 55° and 82°.

Find the measure of the third angle.

Show Solution

Steps	Explanation
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.	[latex]m\angle A + m\angle B = 180[/latex][latex]55+82+x=180[/latex]
Step 5. Solve the equation.	[latex]137+x=180[/latex] [latex]x=43[/latex]
Step 6. Check:	[latex]55+82+43 \stackrel{?}{=} 180[/latex] [latex]✔ \qquad 180 = 180[/latex]
Step 7. Answer the question.	The measure of the third angle is 43 degrees.

Try It

The measures of two angles of a triangle are 31° and 128°.

Find the measure of the third angle.

Show Solution

[latex]21^{\circ}[/latex]

Try It

A triangle has angles of 49° and 75°.

Find the measure of the third angle.

Show Solution

[latex]56^{\circ}[/latex]

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one 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** A right triangle.

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.

Finding Measure of Third Angle in a Triangle

One angle of a right triangle measures 28°.
What is the measure of the third angle?

Show Solution

Steps	Explanation
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 an angle
Step 4. Translate. Write the appropriate formula and substitute.	[latex]m\angle A + m\angle B = 180[/latex][latex]x+90+28=180[/latex]
Step 5. Solve the equation.	[latex]x+118=180[/latex][latex]x=62[/latex]
Step 6. Check:	[latex]90+28+62 \stackrel{?}{=} 180[/latex][latex]✔ \qquad 180 = 180[/latex]
Step 7. Answer the question.	The measure of the third angle is 62 degrees.

Try It

One angle of a right triangle measures 56°.

What is the measure of the other angle?

Show Solution

[latex]34^{\circ}[/latex]

Try It

One angle of a right triangle measures 45°.

What is the measure of the other angle?

Show Solution

[latex]45^{\circ}[/latex]

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.

Finding the Measures of Angles in a Triangle

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

Show Solution

Steps	Explanation
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 = angle 1 a+20 = angle 2
Step 4. Translate. Write the appropriate formula and substitute into the formula.	[latex]m\angle A + m\angle B = 180[/latex] [latex]a+(a+20)+90=180[/latex]
Step 5. Solve the equation.	[latex]2a+110=180[/latex] [latex]2a=70[/latex] [latex]a=35[/latex] (angle 1) angle 2 [latex]= a+20 = 35+20=55[/latex] angle 3 = 90 degrees, the right angle
Step 6. Check:	[latex]35+55+90 \stackrel{?}{=} 180[/latex] [latex]✔ \qquad 180 = 180[/latex]
Step 7. Answer the question.	The three angles measure 35°, 55°, and 90°.

Try It

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

Show Solution

[latex]20^{\circ}, 70^{\circ}, 90^{\circ}[/latex]

Try It

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

Show Solution

[latex]30^{\circ}, 60^{\circ}, 90^{\circ}[/latex]

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 7 are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.

Similar triangles. The larger has sides 12, 16 and 20. The smaller has sides 3,4, and 5. Below is the proportion 16/4 =20/5 =12/3. — **Figure 7** Similar triangles.

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.

description in caption — Two similar triangles are shown. The larger triangle is on the left and is labeled with vertices A, B, and C Side a is across from angle A, side b is across from angle B, and side c is across from angle C. The smaller triangle on the right is labeled with vertices X, Y, and Z. Side x is across from angle X, side y is across from angle Y, and side z is across from angle Z. Several equivalences are listed: The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. The proportion a divided by x is equal to the proportion b divided by y which is equal to the proportion c divided by z.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in ΔABC:

$\begin{matrix} the length a can also be written B C \\ the length b can also be written A C \\ the length c can also be written A B \end{matrix}$

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

Finding Side Lengths in Similar Triangles

ΔABC and ΔXYZ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Show Solution

Solution

Steps	Explanation
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 ΔABC
	y = length of the third side of ΔXYZ
Step 4. Translate.	The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z}$ Since the side AB=4 corresponds to the side XY=3, we will use the ratio [latex]\frac{AB}{XY}=\frac{4}{3}[/latex] to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equations.
Step 6. Check:
Step 7. Answer the question.	The third side of ΔABC is 6 and the third side of ΔXYZ is 2.4.

Try It

ΔABC is similar to ΔXYZ. Find a.

Pair of similar triangles. The first has sides 15, 17, and a. The second has sides y, 25.5, and 12.

Show Solution

8

Try It

ΔABC is similar to ΔXYZ. Find y.

Show Solution

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 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 8.

Three right triangles. The two sides touching the right angle are the “leg” in each. The sides across from the right angles are “hypotenuse.” — **Figure 8** Right triangles.

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 ΔABC,

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

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

To solve problems that use the Pythagorean Theorem, we will need to find square roots. We use the notation [latex]\sqrt{m}[/latex] and define it in this way:

If [latex]m=n^2[/latex], then [latex]\sqrt{m}=n[/latex] for [latex]n \geq 0[/latex].

For example, [latex]\sqrt{25}[/latex] is 5 because [latex]5^2=25[/latex].

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

Using Pythagorean Theorem to Find Length of Hypotenuse

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

Show Solution

Right triangle with legs labeled 3 and 4.

Steps	Explanation
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= the length of the hypotenuse
Step 4. Translate. Write the appropriate formula. Substitute.	[latex]a^2+b^2=c^2[/latex] [latex]3^2+4^2=c^2[/latex]
Step 5. Solve the equation.	[latex]9+16=c^2[/latex] [latex]25=c^2[/latex] [latex]\sqrt{25}=c[/latex] [latex]5=c[/latex]
Step 6. Check:	[latex]3^2+4^2\stackrel{?}{=}5^2[/latex] [latex]9+16\stackrel{?}{=}25[/latex] [latex]✔ \qquad 25 = 25[/latex]
Step 7. Answer the question.	The length of the hypotenuse is 5.

Try It

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

Show Solution

10

Try It

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

Show Solution

17

Using Pythagorean Theorem to Find Unknown Side Length

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

Show Solution

Steps	Explanation
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=the leg of the triangle Label side b
Step 4. Translate. Write the appropriate formula. Substitute.	[latex]a^2+b^2=c^2[/latex] [latex]5^2+b^2=13^2[/latex]
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify.	[latex]25+b^2=169[/latex] [latex]b^2=144[/latex] [latex]b= \sqrt{144}[/latex] [latex]b=12[/latex]
Step 6. Check:	[latex]5^2+12^2 \stackrel{?}{=} 13^2[/latex] [latex]25+144 \stackrel{?}{=} 169[/latex] [latex]✔ \qquad 169 = 169[/latex]
Step 7. Answer the question.	The length of the leg is 12.

Try It

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

Show Solution

8

Try It

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

Show Solution

12

Using Pythagorean Theorem to Find Unknown Side Length

Kelvin is building a gazebo and wants to brace each corner by placing a 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.

Show Solution

Steps	Explanation
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.	[latex]a^2+b^2=c^2[/latex][latex]x^2+x^2=10^2[/latex]
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.	[latex]2x^2=100[/latex] [latex]x^2=50[/latex] [latex]\sqrt{x^2}=x= \sqrt{50}[/latex] [latex]b \approx 7.1[/latex]
Step 6. Check:	[latex]a^2+b^2 \stackrel{?}{=} c^2[/latex] [latex]7.1^2+7.1^2 \stackrel{?}{=} 10^2[/latex] ✔
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Try It

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

Show Solution

12 feet

Try It

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

Show Solution

8 feet

Key Concepts

Supplementary and Complementary Angles

If the sum of the measures of two angles is [latex]180^\circ[/latex], then the angles are supplementary.
If [latex]\angle A[/latex] and [latex]\angle B[/latex] are supplementary, then [latex]m\angle A+m\angle B=180^\circ[/latex].
If the sum of the measures of two angles is [latex]90^\circ[/latex], then the angles are complementary.
If [latex]\angle A[/latex] and [latex]\angle B[/latex] are complementary, then [latex]m\angle A+m\angle B=90^\circ[/latex].

Solve 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.

Sum of the Measures of the Angles of a Triangle

A triangle with vertices labeled A, B, and C.

For any $Δ A B C,$ the sum of the measures is 180°
$m ∠ A + m ∠ B + m ∠ C = 180$

Right Triangle

A 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.

Section Exercises

Use the Properties of Angles

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

1. 53°

Show Solution

(a) 127°
(b) 37°

2. 16°

3. 29°

Show Solution

(a) 151°
(b) 61°

4. 72°

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

5. Find the supplement of a $135°$ angle.

Show Solution

45°

6. Find the complement of a $38°$ angle.

7. Find the complement of a $27.5°$ angle.

Show Solution

62.5°

8. Find the supplement of a $109.5°$ angle.

9. Two angles are supplementary. The larger angle is $56°$ more than the smaller angle. Find the measures of both angles.

Show Solution

62°, 118°

10. Two angles are supplementary. The smaller angle is $36°$ less than the larger angle. Find the measures of both angles.

11. Two angles are complementary. The smaller angle is $34°$ less than the larger angle. Find the measures of both angles.

Show Solution

62°, 28°

12. Two angles are complementary. The larger angle is $52°$ 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 $26°$ and $98° .$ Find the measure of the third angle.

Show Solution

56°

14. The measures of two angles of a triangle are $61°$ and $84° .$ Find the measure of the third angle.

15. The measures of two angles of a triangle are $105°$ and $31° .$ Find the measure of the third angle.

Show Solution

44°

16. The measures of two angles of a triangle are $47°$ and $72° .$ Find the measure of the third angle.

17. One angle of a right triangle measures $33° .$ What is the measure of the other angle?

Show Solution

57°

18. One angle of a right triangle measures $51° .$ 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?

Show Solution

67.5°

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.

Show Solution

45°, 45°, 90°

22. The measure of the smallest angle of a right triangle is $20°$ 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.

Show Solution

30°, 60°, 90°

24. The angles in a triangle are such that the measure of one angle is $20°$ 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, $Δ A B C$ is similar to $Δ X Y Z .$ Find the length of the indicated side.

Pair of similar triangles. The left triangle has sides 9, b, and 15. The side across from X is x, the side across from Y is 8, and the side across from Z is 10.

25. side $b$

Show Solution

12

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 with side across from San Francisco is 1 inch, the side across from Las Vegas is 1.3 inches, and the side across from Los Angeles is 2.1 inches.

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

Show Solution

351 miles

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.

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.

Show Solution

15

30.

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.

31.

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.

Show Solution

25

32.

A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.

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.

A right with side across from the right angle of 10. One of the sides touching the right angle is labeled as 6.

Show Solution

8

34.

A right triangle is shown. The side across from the right angle is 17. One of the sides touching the right angle is 8.

35.

A right triangle is shown. The side across from the right angle is 13. One of the sides touching the right angle is 5.

Show Solution

12

36.

A right triangle is shown. The side across from the right angle is 20. One of the sides touching the right angle is 16.

37.

A right triangle is shown. The side across from the right angle is 13. One of the sides touching the right angle is 8.

Show Solution

10.2

38.

A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.

39.

A right triangle is shown. The side across from the right angle is 17. One of the sides touching the right angle is 15.

Show Solution

8

40.

A right triangle is shown. The side across from the right angle is 7. One of the sides touching the right angle is 5.

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

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

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

Show Solution

5 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 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 has a side is of 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.

Show Solution

14.1 feet

44. Brian borrowed a $20-foot$ 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 house with a 20 foot ladder leaning against it. The horizontal distance from the house to the base of the ladder is 6 feet.

Everyday Math

45. Building a scale model Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is $30$ feet wide and $35$ feet tall at the highest point of the roof. If the dollhouse will be $2.5$ feet wide, how tall will its highest point be?

Show Solution

2.9 feet

46. Measurement A city engineer plans to build a footbridge across a lake from point $X$ to point $Y,$ as shown in the picture below. To find the length of the footbridge, she draws a right triangle $XYZ,$ with right angle at $X .$ She measures the distance from $X$ to $Z, 800$ feet, and from $Y$ to $Z, 1,000$ feet. How long will the bridge be?

A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.

Writing Exercises

47. Write three of the properties of triangles from this section and then explain each in your own words.

48. Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length $3$ and $4 .$

Three squares are shown, illustrating the areas involved in the Pythagorean Theorem.

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 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 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 180°, then they are called supplementary angles.

vertex of an angle

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

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1.1 Use Properties of Angles, Triangles, and the Pythagorean Theorem Copyright © by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

Use the Properties of Angles

How To

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Use the Properties of Triangles

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Right Triangles

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Similar Triangles

Solution

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Use the Pythagorean Theorem

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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

Everyday Math

Writing Exercises

GLOSSARY

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