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

Learning objectives

Learning objectives

• 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 $180\mathrm{'?}$ 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 9.5.

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 9.6, $∠A$ is the angle with vertex at point $A.$ The measure of $∠A$ is written $m\mathrm{∠}A.$

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 $∠A$ is $\mathrm{27^{\circ}},$ we would write $m\mathrm{∠}A=27.$

If the sum of the measures of two angles is $\mathrm{180^{\circ}},$ then they are called supplementary angles. In Figure 9.7, each pair of angles is supplementary because their measures add to $\mathrm{180^{\circ}}.$ Each angle is the supplement of the other.

If the sum of the measures of two angles is $\mathrm{90^{\circ}},$ then the angles are complementary angles. In Figure 9.8, each pair of angles is complementary, because their measures add to $\mathrm{90^{\circ}}.$ Each angle is the complement of the other.

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve 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.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

Example 9.16

Solution

1. Find the supplement. A supplementary angle makes a total angle of $180^{\\circ}$. Let $s$ be the supplement of the $40^{\\circ}$ angle, so $s+40=180$. Subtracting $40$ gives $s=140$. The figures show the straight-line diagram, this equation, and the check $140+40=180$.
2. Find the complement. A complementary angle makes a total angle of $90^{\\circ}$. Let $c$ be the complement of the $40^{\\circ}$ angle, so $c+40=90$. Subtracting $40$ gives $c=50$.
3. Check and state both measures. The complement figures confirm that $50+40=90$, so ⓐ the supplement is $140^{\\circ}$ and ⓑ the complement is $50^{\\circ}$.

Takeaway: Supplementary angles sum to $180^{\\circ}$, whereas complementary angles sum to $90^{\\circ}$.

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

Example 9.17

Solution

1. Express the angle measures. Let $a$ be the measure of the smaller angle in degrees. The larger angle measures $a+30$ degrees, as the first figures show.
2. Use the supplementary-angle property. The two angles satisfy $a+(a+30)=180$. Combining like terms gives $2a+30=180$, matching the figures that display the equation.
3. Solve for the smaller angle. Subtracting $30$ gives $2a=150$, and dividing by $2$ gives $a=75$. The larger angle is $a+30=75+30=105$ degrees.
4. Check the measures. The figures verify that $75+105=180$ and that the larger angle is $30^{\\circ}$ greater. The measures of the angles are $75^{\\circ}$ and $105^{\\circ}$.

Takeaway: For supplementary angles, use the $180^{\\circ}$ sum to form the equation and then verify the stated difference.

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 9.9 is called $\Delta{}ABC,$ read ‘triangle $\mathrm{ABC}$’. 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 $\mathrm{180^{\circ}}.$

Example 9.18

Solution

1. Write the angle-sum equation. Let $x$ be the measure in degrees of the third angle. The triangle-angle property gives $55+82+x=180$, which is shown in the figures with the triangle labeled $55^{\\circ}$, $82^{\\circ}$, and $x$.
2. Combine the known angles. Since $55+82=137$, the equation becomes $137+x=180$.
3. Solve for the unknown angle. Subtracting $137$ from both sides gives $x=43$.
4. Check the angle sum. The figures verify $55+82+43=180$, so the measure of the third angle is $43^{\\circ}$.

Takeaway: The three interior angles of every triangle total $180^{\\circ}$, regardless of the triangle's shape.

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one $\mathrm{90^{\circ}}$ angle, which is often marked with the symbol shown in Figure 9.10.

If we know that a triangle is a right triangle, we know that one angle measures $\mathrm{90^{\circ}}$ so we only need the measure of one of the other angles in order to determine the measure of the third angle.

Example 9.19

Solution

1. Identify all three angles. A right triangle has one $90^{\\circ}$ angle, and another angle measures $28^{\\circ}$. Let $x$ be the third angle's measure in degrees, as the triangle diagram shows.
2. Apply the triangle angle sum. The angles satisfy $x+90+28=180$, and combining the known measures gives $x+118=180$.
3. Solve for the third angle. Subtracting $118$ from both sides gives $x=62$. The figures display the equation and the result.
4. Check the angle sum. Since $90+28+62=180$, the third angle measures $62^{\\circ}$.

Takeaway: In a right triangle, the two acute angles together measure $90^{\\circ}$ because the full angle sum is $180^{\\circ}$.

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 9.20

Solution

1. Express the acute angle measures. Let $a$ be the measure in degrees of the smallest angle. The other acute angle is $a+20$ degrees, and the right angle is $90^{\\circ}$. The figures label all three angles in this way.
2. Form the triangle angle equation. The sum of the angles gives $a+(a+20)+90=180$. Combining like terms produces $2a+110=180$, as shown in the figures.
3. Solve for the smallest angle. Subtracting $110$ gives $2a=70$, and dividing by $2$ gives $a=35$. The second acute angle is $35+20=55$ degrees.
4. Check the three measures. The figures verify that $35+55+90=180$. The angles measure $35^{\\circ}$, $55^{\\circ}$, and $90^{\\circ}$.

Takeaway: The two acute angles of a right triangle total $90^{\\circ}$, which provides a quick check on their measures.

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 9.11 are similar. Each side of $\Delta{}ABC$ is four times the length of the corresponding side of $\Delta{}XYZ$ and their corresponding angles have equal measures.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $\Delta{}ABC:$

$\,\begin{array}{l} \mathrm{the length}\,a\,\mathrm{can also be written}\,BC \\ \mathrm{the length}\,b\,\mathrm{can also be written}\,AC \\ \mathrm{the length}\,c\,\mathrm{can also be written}\,AB \\ \end{array}$

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

Example 9.21

Solution

1. Set up corresponding-side proportions. From the similar triangles, the ratio of corresponding sides is $\\frac{4}{3}$. For the larger triangle, the proportion is $\\frac{4}{3}=\\frac{a}{4.5}$; for the smaller triangle, it is $\\frac{4}{3}=\\frac{3.2}{y}$. The first figure displays these ratios.
2. Solve for the larger missing side. Cross-multiplication gives $3a=4(4.5)=18$, so $a=6$. The second figure shows this calculation.
3. Solve for the smaller missing side. Cross-multiplication gives $4y=3(3.2)=9.6$, so $y=2.4$.
4. Check the proportions. The final figure verifies both cross-products: $4(4.5)=3(6)$ and $4(2.4)=3(3.2)$. The missing side lengths are $6$ and $2.4$.

Takeaway: In similar triangles, match corresponding sides before cross-multiplying a proportion for a missing length.

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 $\mathrm{90^{\circ}}$ angle, which we usually mark with a small square in the corner. The side of the triangle opposite the $\mathrm{90^{\circ}}$ angle is called the hypotenuse, and the other two sides are called the legs. See Figure 9.12.

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:

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 9.22

Solution

1. Identify the side roles. The triangle has legs of lengths $3$ and $4$, and the hypotenuse is $c$. The diagram shows the $3$ - $4$ - $5$ right triangle.
2. Apply the Pythagorean Theorem. The formula is $a^2+b^2=c^2$, so substituting gives $3^2+4^2=c^2$.
3. Evaluate and solve. Squaring the legs gives $9+16=c^2$, hence $25=c^2$. Taking the positive square root gives $c=5$. The calculation figures show these steps.
4. Check the result. Since $3^2+4^2=9+16=25=5^2$, the hypotenuse is $5$ units long.

Takeaway: The hypotenuse is the side opposite the right angle and must be the unknown in $a^2+b^2=c^2$.

Example 9.23

Solution

1. Identify the missing leg. The right triangle has one leg of length $5$, the other leg $b$, and hypotenuse $13$. The diagram marks the right angle between the legs.
2. Apply the Pythagorean Theorem. Using $a^2+b^2=c^2$, substitute the known lengths to get $5^2+b^2=13^2$.
3. Solve for the missing length. Simplifying gives $25+b^2=169$, so $b^2=144$. Taking the positive square root gives $b=12$, as the solution figure shows.
4. Check the result. Since $5^2+12^2=25+144=169=13^2$, the longer leg is $12$ units.

Takeaway: When a leg is unknown, subtract the known leg's square from the hypotenuse's square before taking a square root.

Example 9.24

Solution

1. Identify the equal legs. The bracket is the hypotenuse, measuring $10$ inches. The distances from the corner to its ends are equal, so each leg has length $x$ inches, as the diagram shows.
2. Apply the Pythagorean Theorem. Since $a^2+b^2=c^2$, substituting equal legs and the hypotenuse gives $x^2+x^2=10^2$.
3. Solve for the distance below the corner. Combining terms gives $2x^2=100$, so $x^2=50$ and $x=\\sqrt{50}\\approx7.1$. The calculation figures show these steps.
4. Check the approximation. Since $(7.1)^2+(7.1)^2\\approx100=10^2$, Kelvin should fasten the bracket approximately $7.1$ inches below the corner.

Takeaway: For an isosceles right triangle with hypotenuse $c$, each leg is $\\frac{c}{\\sqrt{2}}$.

Section 9.3 Exercises

Section 9.3 Exercises

Use the Properties of Angles

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

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

Use the Properties of Triangles

In the following exercises, solve using properties of triangles.

Find the Length of the Missing Side

In the following exercises, $\Delta{}ABC$ is similar to $\Delta{}XYZ.$ Find the length of the indicated side.

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.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

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.

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

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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

2. 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.

1. An angle measures $\mathrm{25^{\circ}}.$ Find its: ⓐ supplement ⓑ complement.

2. An angle measures $\mathrm{77^{\circ}}.$ Find its: ⓐ supplement ⓑ complement.

1. Two angles are supplementary. The larger angle is $\mathrm{100^{\circ}}$ more than the smaller angle. Find the measures of both angles.

2. Two angles are complementary. The larger angle is $\mathrm{40^{\circ}}$ more than the smaller angle. Find the measures of both angles.

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

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

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

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

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

2. The measure of one angle of a right triangle is $\mathrm{30^{\circ}}$ more than the measure of the smallest angle. Find the measures of all three angles.

1. John puts the base of a $\mathrm{13-ft}$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

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

1. $\mathrm{53^{\circ}}$

2. $\mathrm{29^{\circ}}$

1. Find the supplement of a $\mathrm{135^{\circ}}$ angle.

2. Find the complement of a $27.5^{\circ}$ angle.

1. Two angles are supplementary. The larger angle is $\mathrm{56^{\circ}}$ more than the smaller angle. Find the measures of both angles.

2. Two angles are complementary. The smaller angle is $\mathrm{34^{\circ}}$ less than the larger angle. Find the measures of both angles.

1. The measures of two angles of a triangle are $\mathrm{26^{\circ}}$ and $\mathrm{98^{\circ}}.$ Find the measure of the third angle.

2. The measures of two angles of a triangle are $\mathrm{105^{\circ}}$ and $\mathrm{31^{\circ}}.$ Find the measure of the third angle.

1. One angle of a right triangle measures $\mathrm{33^{\circ}}.$ What is the measure of the other angle?

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

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

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

1. 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?

2. 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?

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