Use Properties of Angles, Triangles, and 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.

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.

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 $50^\circ$.

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:

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 $105^\circ$.

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 $180^\circ$.

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:

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

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:

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

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:

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 $90^\circ$.

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 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:

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

To find $y$:

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 $2.4$.

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.

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:

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

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:

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

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:

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

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.

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, the house and sailboat illustrations, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.