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Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use the properties of angles

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 .

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 , \(\text{∠}A\) is the angle with vertex at point \(A.\) The measure of \(\text{∠}A\) is written \(m∠A.\)

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

If the sum of the measures of two angles is \(\text{180^{\circ}},\) then they are called supplementary angles. In , each pair of angles is supplementary because their measures add to \(\text{180^{\circ}}.\) Each angle is the supplement of the other.

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

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

The three angles of a triangle are related in a special way. The sum of their measures is \(\text{180^{\circ}}.\)

\[m\text{∠}A+m\text{∠}B+m\text{∠}C=\text{180^{\circ}}\]

Example

Try it.

The measures of two angles of a triangle are \(\text{55^{\circ}}\) and \(\text{82^{\circ}}.\) 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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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 \(\text{90^{\circ}}\) angle, which we usually mark with a small square in the corner. The side of the triangle opposite the \(\text{90^{\circ}}\) 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 \(\sqrt{m}\) and defined it in this way:

\[\text{If}\ m={n}^{2},\ \text{then}\ \sqrt{m}=n\ \text{for}\ n\ge 0\]

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

Try it.

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.

Example

Try it.

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Supplementary and Complementary Angles
- If the sum of the measures of two angles is 180°, then the angles are supplementary.
- If \(\text{∠}A\) and \(\text{∠}B\) are supplementary, then \(m\text{∠}A+m\text{∠}B=180\).
- If the sum of the measures of two angles is 90°, then the angles are complementary.
- If \(\text{∠}A\) and \(\text{∠}B\) are complementary, then \(m\text{∠}A+m\text{∠}B=90\).
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 \(\text{\Delta }ABC,\) the sum of the measures is 180°
- \(m\text{∠}A+m\text{∠}B+m\text{∠}C=180\)
Right Triangle
- A right triangle is a triangle that has one 90° angle, which is often marked with a \(\text{⦜}\)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.

Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use the Properties of Angles

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

Try it.

\(\text{53^{\circ}}\)

Solution

ⓐ 127°
ⓑ 37°

Try it.

\(\text{16^{\circ}}\)

Try it.

\(\text{29^{\circ}}\)

Solution

ⓐ 151°
ⓑ 61°

Try it.

\(\text{72^{\circ}}\)

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

Try it.

Find the supplement of a \(\text{135^{\circ}}\) angle.

Solution

45°

Try it.

Find the complement of a \(\text{38^{\circ}}\) angle.

Try it.

Find the complement of a \(27.5^{\circ}\) angle.

Solution

62.5°

Try it.

Find the supplement of a \(109.5^{\circ}\) angle.

Try it.

Two angles are supplementary. The larger angle is \(\text{56^{\circ}}\) more than the smaller angle. Find the measures of both angles.

Solution

62°, 118°

Try it.

Two angles are supplementary. The smaller angle is \(\text{36^{\circ}}\) less than the larger angle. Find the measures of both angles.

Try it.

Two angles are complementary. The smaller angle is \(\text{34^{\circ}}\) less than the larger angle. Find the measures of both angles.

Solution

62°, 28°

Try it.

Two angles are complementary. The larger angle is \(\text{52^{\circ}}\) 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.

Try it.

The measures of two angles of a triangle are \(\text{26^{\circ}}\) and \(\text{98^{\circ}}.\) Find the measure of the third angle.

Solution

56°

Try it.

The measures of two angles of a triangle are \(\text{61^{\circ}}\) and \(\text{84^{\circ}}.\) Find the measure of the third angle.

Try it.

The measures of two angles of a triangle are \(\text{105^{\circ}}\) and \(\text{31^{\circ}}.\) Find the measure of the third angle.

Solution

44°

Try it.

The measures of two angles of a triangle are \(\text{47^{\circ}}\) and \(\text{72^{\circ}}.\) Find the measure of the third angle.

Try it.

One angle of a right triangle measures \(\text{33^{\circ}}.\) What is the measure of the other angle?

Solution

57°

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One angle of a right triangle measures \(\text{51^{\circ}}.\) What is the measure of the other angle?

Try it.

One angle of a right triangle measures \(22.5^{\circ}.\) What is the measure of the other angle?

Solution

67.5°

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One angle of a right triangle measures \(36.5^{\circ}.\) What is the measure of the other angle?

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The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

Solution

45°, 45°, 90°

Try it.

The measure of the smallest angle of a right triangle is \(\text{20^{\circ}}\) less than the measure of the other small angle. Find the measures of all three angles.

Try it.

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.

Solution

30°, 60°, 90°

Try it.

The angles in a triangle are such that the measure of one angle is \(\text{20^{\circ}}\) 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, \(\text{\Delta }ABC\) is similar to \(\text{\Delta }XYZ.\) Find the length of the indicated side.

Try it.

side \(b\)

Solution

Try it.

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.

Try it.

Find the distance from Los Angeles to San Francisco.

Solution

351 miles

Try it.

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.

Try it.

Solution

Try it.

Solution

Try it.

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.

Try it.

Solution

Try it.

Solution

Try it.

Solution

10.2

Try it.

Solution

16.2

Try it.

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

Try it.

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?

Solution

5 feet

Try it.

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?

Try it.

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?

Solution

14.1 feet

Try it.

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?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Solve: \(x+3+6=11.\)
If you missed this problem, review .

Afslør svaret

\(x=2\)
Solve: \(\frac{a}{45}=\frac{4}{3}.\)
If you missed this problem, review .

Afslør svaret

\(60\)
Simplify: \(\sqrt{36+64}.\)
If you missed this problem, review .

Afslør svaret

\(10\)

An angle measures \(\text{40^{\circ}}.\) Find ⓐ its supplement, and ⓑ its complement.

Afslør svaret

ⓐ
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 for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

ⓑ
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 for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

An angle measures \(\text{25^{\circ}}.\) Find its: ⓐ supplement ⓑ complement.
Afslør svaret
1. ⓐ 155°
2. ⓑ 65°
An angle measures \(\text{77^{\circ}}.\) Find its: ⓐ supplement ⓑ complement.
Afslør svaret
1. ⓐ 103°
2. ⓑ 13°

Two angles are supplementary. The larger angle is \(\text{30^{\circ}}\) more than the smaller angle. Find the measure of both angles.

Afslør svaret

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. The larger angle is 30° more than the smaller angle.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

Two angles are supplementary. The larger angle is \(\text{100^{\circ}}\) more than the smaller angle. Find the measures of both angles.

Afslør svaret

40°, 140°
Two angles are complementary. The larger angle is \(\text{40^{\circ}}\) more than the smaller angle. Find the measures of both angles.

Afslør svaret

25°, 65°

The measures of two angles of a triangle are \(\text{55^{\circ}}\) and \(\text{82^{\circ}}.\) Find the measure of the third angle.

Afslør svaret

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.

The measures of two angles of a triangle are \(\text{31^{\circ}}\) and \(\text{128^{\circ}}.\) Find the measure of the third angle.

Afslør svaret

21°
A triangle has angles of \(\text{49^{\circ}}\) and \(\text{75^{\circ}}.\) Find the measure of the third angle.

Afslør svaret

56°

One angle of a right triangle measures \(\text{28^{\circ}}.\) What is the measure of the third angle?

Afslør svaret

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.

One angle of a right triangle measures \(\text{56^{\circ}}.\) What is the measure of the other angle?

Afslør svaret

34°
One angle of a right triangle measures \(\text{45^{\circ}}.\) What is the measure of the other angle?

Afslør svaret

45°

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

Afslør svaret

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.

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

Afslør svaret

20°, 70°, 90°
The measure of one angle of a right triangle is \(\text{30^{\circ}}\) more than the measure of the smallest angle. Find the measures of all three angles.

Afslør svaret

30°, 60°, 90°

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

Afslør svaret

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 \(\frac{\text{AB}}{\text{XY}}=\frac{4}{3}\) 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.

\(\text{\Delta }ABC\) is similar to \(\text{\Delta }XYZ.\) Find \(a.\)

Afslør svaret

8
\(\text{\Delta }ABC\) is similar to \(\text{\Delta }XYZ.\) Find \(y.\)

Afslør svaret

22.5

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

Afslør svaret

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.

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

Afslør svaret

10
Use the Pythagorean Theorem to find the length of the hypotenuse.

Afslør svaret

17

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

Afslør svaret

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.

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

Afslør svaret

8
Use the Pythagorean Theorem to find the length of the leg.

Afslør svaret

12

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.

Afslør svaret

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.

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?

Afslør svaret

12 feet
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?

Afslør svaret

8 feet
\(\text{53^{\circ}}\)
Afslør svaret
1. ⓐ 127°
2. ⓑ 37°
\(\text{16^{\circ}}\)
\(\text{29^{\circ}}\)
Afslør svaret
1. ⓐ 151°
2. ⓑ 61°
\(\text{72^{\circ}}\)
Find the supplement of a \(\text{135^{\circ}}\) angle.

Afslør svaret

45°
Find the complement of a \(\text{38^{\circ}}\) angle.
Find the complement of a \(27.5^{\circ}\) angle.

Afslør svaret

62.5°
Find the supplement of a \(109.5^{\circ}\) angle.
Two angles are supplementary. The larger angle is \(\text{56^{\circ}}\) more than the smaller angle. Find the measures of both angles.

Afslør svaret

62°, 118°
Two angles are supplementary. The smaller angle is \(\text{36^{\circ}}\) less than the larger angle. Find the measures of both angles.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\approx

approximately equal
Equal to the precision shown, not exactly.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use the properties of angles
Use the properties of triangles
Use the Pythagorean Theorem
If the sum of the measures of two angles is 180°, then the angles are supplementary.
If
If the sum of the measures of two angles is 90°, then the angles are complementary.
If
Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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