Appendix L: Triangles and the Pythagorean Theorem

Izabela Mazur

Appendix L: Triangles and the Pythagorean Theorem

Izabela Mazur

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 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.
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 3.1

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

Show answer

21°

TRY IT 3.2

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

Show 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.
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 4.1

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

Show answer

34°

TRY IT 4.2

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

Show 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.
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 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.

Show 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.

Show answer

30°, 60°, 90°

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.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.

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.

Right triangle with legs labeled as 3 and 4.

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 7.1

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

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.

Show answer

10

TRY IT 7.2

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

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.

Show answer

17

EXAMPLE 8

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

Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.

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 8.1

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

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.

Show answer

8

TRY IT 8.2

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

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.

Show 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.
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 9.1

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.

Show 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.

Show answer

8 feet

Summary

Sum of the Measures of the Angles of a Triangle
- For any $\Delta ABC$ , the sum of the measures is 180°
- $m\angle A+m\angle B=180$
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.

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

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Physics 131: What Is Physics? by Izabela Mazur is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Use the Properties of Triangles

Right Triangles

Use the Pythagorean Theorem

Summary

Attributions

License

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