Use Properties of Angles, Triangles, and the Pythagorean Theorem

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. In this section, we apply our problem-solving strategy 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.

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

We measure angles in degrees, using the symbol ° to represent degrees, and the abbreviation $m$ for the measure of an angle. So if $\angle A$ is $27^\circ$ , we write $m\angle A = 27$ .

If the sum of the measures of two angles is $180^\circ$ , the angles are called supplementary angles — each angle is the supplement of the other. If the sum of the measures of two angles is $90^\circ$ , the angles are called complementary angles — each angle is the complement of the other.

Supplementary and complementary angles. If the sum of the measures of two angles is $180^\circ$ , the angles are supplementary. If $\angle A$ and $\angle B$ are supplementary, then $m\angle A + m\angle B = 180^\circ$ .

If the sum of the measures of two angles is $90^\circ$ , the angles are complementary. If $\angle A$ and $\angle B$ are complementary, then $m\angle A + m\angle B = 90^\circ$ .

In this section and the next, geometry formulas will name the variables and give us the equation to solve. Since these applications all involve geometric shapes, it will also help to draw a figure and label it with the information from the problem.

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.

Example. An angle measures $40^\circ$ . Find (a) its supplement, and (b) its complement.

(a) Let $s =$ the measure of the supplement. Since supplementary angles sum to $180^\circ$ : $s + 40 = 180$ , so $s = 140$ . Checking: $140 + 40 = 180$ . The supplement of the $40^\circ$ angle is $140^\circ$ .

(b) Let $c =$ the measure of the complement. Since complementary angles sum to $90^\circ$ : $c + 40 = 90$ , so $c = 50$ . Checking: $50 + 40 = 90$ . The complement of the $40^\circ$ angle is $50^\circ$ .

An angle measures $25^\circ$ . Find its supplement.

An angle measures $77^\circ$ . Find its complement.

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

Example. Two angles are supplementary. The larger angle is $30^\circ$ more than the smaller angle. Find the measure of both angles.

Let $a =$ measure of the smaller angle, so $a + 30 =$ measure of the larger angle. Since the angles are supplementary:

(a + 30) + a = 180

Combining like terms: $2a + 30 = 180$ , so $2a = 150$ and $a = 75$ (the smaller angle). The larger angle is $a + 30 = 105$ . Checking: $75 + 105 = 180$ . The measures of the angles are $75^\circ$ and $105^\circ$ .

Two angles are supplementary. The larger angle is $100^\circ$ more than the smaller angle. Find the measure of the smaller angle.

Two angles are complementary. The larger angle is $40^\circ$ more than the smaller angle. Find the measure of the smaller angle.

Use the properties of triangles

Triangles have three sides and three angles, and are named by their vertices. The triangle below is called $\Delta ABC$ , read “triangle ABC.” Each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

The three angles of a triangle are related in a special way: the sum of their measures is $180^\circ$ .

Sum of the measures of the angles of a triangle. For any $\Delta ABC$ , the sum of the measures of the angles is $180^\circ$ :

m\angle A + m\angle B + m\angle C = 180^\circ

Example. The measures of two angles of a triangle are $55^\circ$ and $82^\circ$ . Find the measure of the third angle.

Let $x =$ the measure of the third angle. Substituting into the angle-sum formula:

55 + 82 + x = 180 \qquad\Rightarrow\qquad 137 + x = 180 \qquad\Rightarrow\qquad x = 43

Checking: $55 + 82 + 43 = 180$ . The measure of the third angle is $43$ degrees.

The measures of two angles of a triangle are $31^\circ$ and $128^\circ$ . Find the measure of the third angle.

A triangle has angles of $49^\circ$ and $75^\circ$ . Find the measure of the third angle.

Right triangles

A right triangle has one $90^\circ$ angle, often marked with a small square in the corner.

If we know a triangle is a right triangle, one angle measures $90^\circ$ , so we only need the measure of one of the other angles to find the third.

Example. One angle of a right triangle measures $28^\circ$ . What is the measure of the third angle?

Let $x =$ the measure of the third angle:

x + 90 + 28 = 180 \qquad\Rightarrow\qquad x + 118 = 180 \qquad\Rightarrow\qquad x = 62

Checking: $90 + 28 + 62 = 180$ . The measure of the third angle is $62^\circ$ .

One angle of a right triangle measures $56^\circ$ . What is the measure of the other angle?

One angle of a right triangle measures $45^\circ$ . What is the measure of the other angle?

When one angle is defined in terms of another, it helps to write expressions for all the angles before drawing the figure.

Example. The measure of one angle of a right triangle is $20^\circ$ more than the measure of the smallest angle. Find the measures of all three angles.

Let $a =$ the first (smallest) angle, so $a + 20 =$ the second angle, and $90 =$ the third angle (the right angle). Substituting into the angle-sum formula:

a + (a + 20) + 90 = 180 \qquad\Rightarrow\qquad 2a + 110 = 180

2a = 70 \qquad\Rightarrow\qquad a = 35 \text{ (first angle)}

The second angle is $a + 20 = 55$ , and the third angle is $90$ . Checking: $35 + 55 + 90 = 180$ . The three angles measure $35^\circ$ , $55^\circ$ , and $90^\circ$ .

The measure of one angle of a right triangle is $50^\circ$ more than the measure of the smallest angle. Find the measure of the smallest angle.

The measure of one angle of a right triangle is $30^\circ$ more than the measure of the smallest angle. Find the measure of the smallest angle.

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.

Properties of similar triangles. If two triangles are similar, their corresponding angle measures are equal and their corresponding side lengths are in the same ratio. For $\Delta ABC$ similar to $\Delta XYZ$ :

m\angle A = m\angle X, \quad m\angle B = m\angle Y, \quad m\angle C = m\angle Z

\frac{a}{x} = \frac{b}{y} = \frac{c}{z}

The length of a side of a triangle may also be referred to by its endpoints — two vertices of the triangle. For example, in $\Delta ABC$ , the length $a$ can also be written $BC$ , the length $b$ can also be written $AC$ , and the length $c$ can also be written $AB$ . This notation helps match up corresponding side lengths when solving similar triangles.

Example. $\Delta ABC$ and $\Delta XYZ$ are similar triangles. In $\Delta ABC$ , side $AB = 4$ and side $AC = 3.2$ ; in $\Delta XYZ$ , the corresponding side $XY = 3$ . Find the length of the third side of each triangle (side $BC = a$ in $\Delta ABC$ , and side $XZ = y$ in $\Delta XYZ$ ), given that side $YZ = 4.5$ .

Since the triangles are similar, corresponding sides are in the same ratio:

\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}

Since $AB = 4$ corresponds to $XY = 3$ , we use the ratio $\frac{AB}{XY} = \frac{4}{3}$ to find the other sides. To find $a$ :

\frac{4}{3} = \frac{a}{4.5} \qquad\Rightarrow\qquad 3a = 4(4.5) \qquad\Rightarrow\qquad 3a = 18 \qquad\Rightarrow\qquad a = 6

To find $y$ :

\frac{4}{3} = \frac{3.2}{y} \qquad\Rightarrow\qquad 4y = 3(3.2) \qquad\Rightarrow\qquad 4y = 9.6 \qquad\Rightarrow\qquad y = 2.4

Checking: $4(4.5) = 6(3)$ gives $18 = 18$ , and $4(2.4) = 3.2(3)$ gives $9.6 = 9.6$ . The third side of $\Delta ABC$ is $6$ , and the third side of $\Delta XYZ$ is $2.4$ .

Triangle ABC is similar to triangle XYZ. Side $AB = 17$ corresponds to side $XY = 25.5$ , and side $BC = a$ corresponds to side $YZ = 12$ . Find $a$ .

Using the same similar triangles ( $AB = 17$ corresponds to $XY = 25.5$ ), side $AC = 15$ corresponds to side $XZ = y$ . Find $y$ .

Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times, named after the Greek philosopher and mathematician Pythagoras. In a right triangle, the side opposite the $90^\circ$ angle is called the hypotenuse, and the other two sides are called the legs.

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other: 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 and $a$ and $b$ are the lengths of the legs.

To solve for a side length, recall that if $m = n^2$ , then $\sqrt{m} = n$ for $n \geq 0$ . For example, $\sqrt{25}$ is $5$ because $5^2 = 25$ .

Example. Use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle whose legs measure $3$ and $4$ .

Let $c =$ the length of the hypotenuse:

a^2 + b^2 = c^2 \qquad\Rightarrow\qquad 3^2 + 4^2 = c^2

9 + 16 = c^2 \qquad\Rightarrow\qquad 25 = c^2 \qquad\Rightarrow\qquad \sqrt{25} = c \qquad\Rightarrow\qquad 5 = c

Checking: $3^2 + 4^2 = 5^2$ gives $9 + 16 = 25$ . The length of the hypotenuse is $5$ .

Use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle whose legs measure 6 and 8.

Use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle whose legs measure 15 and 8.

Example. Use the Pythagorean Theorem to find the length of the longer leg of a right triangle whose hypotenuse is $13$ and whose shorter leg is $5$ .

Let $b =$ the unknown leg:

a^2 + b^2 = c^2 \qquad\Rightarrow\qquad 5^2 + b^2 = 13^2

25 + b^2 = 169 \qquad\Rightarrow\qquad b^2 = 144 \qquad\Rightarrow\qquad b = \sqrt{144} \qquad\Rightarrow\qquad b = 12

Checking: $5^2 + 12^2 = 13^2$ gives $25 + 144 = 169$ . The length of the leg is $12$ .

Use the Pythagorean Theorem to find the length of the leg of a right triangle whose hypotenuse is 17 and whose other leg is 15.

Use the Pythagorean Theorem to find the length of the leg of a right triangle whose hypotenuse is 15 and whose other leg is 9.

Example. Kelvin is building a gazebo and wants to brace each corner by placing a $10$ -inch wooden bracket diagonally, as shown, so that the distances from the corner to each end of the bracket are equal. How far below the corner should he fasten the bracket? Approximate to the nearest tenth of an inch.

Let $x =$ the distance from the corner along each side. Since both legs are equal:

a^2 + b^2 = c^2 \qquad\Rightarrow\qquad x^2 + x^2 = 10^2

2x^2 = 100 \qquad\Rightarrow\qquad x^2 = 50 \qquad\Rightarrow\qquad x = \sqrt{50} \approx 7.1

Checking: $(7.1)^2 + (7.1)^2 \approx 10^2$ . Kelvin should fasten each piece of wood approximately $7.1$ inches from the corner.

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?

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

Key terms

angle — a figure formed by two rays sharing a common endpoint (the vertex). supplementary angles — two angles whose measures sum to $180^\circ$ . complementary angles — two angles whose measures sum to $90^\circ$ . right triangle — a triangle with one $90^\circ$ angle. similar triangles — triangles with the same shape but not necessarily the same size, whose corresponding angles are equal and corresponding sides are proportional. hypotenuse — the side of a right triangle opposite the right angle. leg (of a right triangle) — either of the two sides that form the right angle. Pythagorean Theorem — in a right triangle, $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse and $a, b$ are the legs.

Practice

Use the properties of angles

Find the supplement of a $53^\circ$ angle. Give the measure in degrees.

Find the complement of a $53^\circ$ angle. Give the measure in degrees.

Find the supplement of a $135^\circ$ angle. Give the measure in degrees.

Find the complement of a $27.5^\circ$ angle. Give the measure in degrees.

Two angles are supplementary. The larger angle is $56^\circ$ more than the smaller angle. Find the measure of the smaller angle, in degrees.

Two angles are supplementary. The larger angle is $56^\circ$ more than the smaller angle. Find the measure of the larger angle, in degrees.

Use the properties of triangles

The measures of two angles of a triangle are $26^\circ$ and $98^\circ$ . Find the measure of the third angle, in degrees.

The measures of two angles of a triangle are $105^\circ$ and $31^\circ$ . Find the measure of the third angle, in degrees.

One angle of a right triangle measures $33^\circ$ . What is the measure of the other angle, in degrees?

One angle of a right triangle measures $22.5^\circ$ . What is the measure of the other angle, in degrees?

$\Delta ABC$ is similar to $\Delta XYZ$ . In $\Delta ABC$ , side $AB = 15$ , side $BC = 9$ , and side $AC = b$ . In $\Delta XYZ$ , side $XY = 10$ , side $YZ = x$ , and side $XZ = 8$ . Find the length of side $b$ .

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle. On the map, Los Angeles to Las Vegas measures 1 inch, Los Angeles to San Francisco measures 1.3 inches, and San Francisco to Las Vegas measures 2.1 inches. The actual distance from Los Angeles to Las Vegas is 270 miles. Find the actual distance from Los Angeles to San Francisco, in miles.

Joe wants to build a doll house for his daughter that looks just like his house. His house is 30 feet wide and 35 feet tall at the highest point of the roof. If the doll house will be 2.5 feet wide, how tall will its highest point be, in feet? Round to the nearest tenth.

Use the Pythagorean Theorem

A right triangle has legs measuring 9 and 12. Use the Pythagorean Theorem to find the length of the hypotenuse.

In a right triangle, the hypotenuse measures 10 and one leg measures 6. Use the Pythagorean Theorem to find the length of the other leg.

In a right triangle, the hypotenuse measures 13 and one leg measures 8. Use the Pythagorean Theorem to find the length of the other leg. Round to the nearest tenth.

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, in feet, should the end of the string of lights be anchored?

Chi is planning to put a path of paving stones through her flower garden, running diagonally from one corner to the opposite corner. The flower garden is a square with sides of 10 feet. What will the length of the path be, in feet? Round to the nearest tenth.

This section is adapted from Prealgebra 2e, Section 9.3: Use Properties of Angles, Triangles, and the Pythagorean Theorem by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the angle, triangle, and Pythagorean Theorem figures as accessible inline graphics; omitted the Be Prepared quiz, the Media callout, and the house and sailboat illustrations; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, restating their diagrams as prose prompts.

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