3.2: Use Properties of Angles, Triangles, and the Pythagorean Theorem

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Learning Objectives

After completing this section, you should be able to:

Use the properties of angles
Use the properties of triangles
Use the Pythagorean Theorem

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 $180 ’?$

The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees. — Figure 3.2.1

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 3.2.2, $∠ A$ is the angle with vertex at point $A .$ The measure of $∠ A$ is written $m ∠ A .$

The image is an angle made up of two rays. The angle is labeled with letter A. — Figure 3.2.2: $∠ A$ is the angle with vertex at $point A .$

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

If the sum of the measures of two angles is $180°$ , as in Figure 3.2.3, each pair of angles is supplementary because their measures add to $180° .$ Each angle is the supplement of the other.

Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees. — Figure 3.2.3: The sum of the measures of supplementary angles is $180° .$

If the sum of the measures of two angles is $90°$ , as in Figure 3.2.4, each pair of angles is complementary, because their measures add to $90° .$ Each angle is the complement of the other.

Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees. — Figure 3.2.4: The sum of the measures of complementary angles is $90° .$

Supplementary and Complementary Angles

If the sum of the measures of two angles is $180°,$ then the angles are supplementary.

If $∠ A$ and $∠ B$ are supplementary, then $m ∠ A + m ∠ B = 180°.$

If the sum of the measures of two angles is $90°,$ then the angles are complementary.

If $∠ A$ and $∠ B$ are complementary, then $m ∠ A + m ∠ B = 90°.$

In the next few sections, you will be introduced to some common geometry formulas. We will use the Problem Solving Strategy for Geometry Applications below. 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.

How To

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.

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 3.2.1

An angle measures $40° .$ Find ⓐ its supplement, and ⓑ its complement.

Answer

ⓐ
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.	$.$

Your Turn 3.2.1

An angle measures $25° .$ Find its: ⓐ supplement ⓑ complement.

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

Example 3.2.2

Two angles are supplementary. The larger angle is $30°$ more than the smaller angle. Find the measure of both angles.

Answer

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.	$.$

Your Turn 3.2.2

Two angles are supplementary. The larger angle is $100°$ more than the smaller angle. Find the measures of both angles.

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 3.2.5 is called $Δ A B C,$ read ‘triangle $ABC$ ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c. — Figure 3.2.5: $Δ A B C$ has vertices $A, B, and C$ and sides $a, b, and c .$

The three angles of a triangle are related in a special way. The sum of their measures is $180° .$

$m ∠ A + m ∠ B + m ∠ C = 180°$

Sum of the Measures of the Angles of a Triangle

For any $Δ A B C,$ the sum of the measures of the angles is $180° .$

$m ∠ A + m ∠ B + m ∠ C = 180°$

Example 3.2.3

The measures of two angles of a triangle are $55°$ and $82° .$ Find the measure of the third angle.

Answer

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.	$.$

Your Turn 3.2.3

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

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one $90°$ .

A right triangle is shown. The right angle is marked with a box and labeled 90 degrees. — Figure 3.2.6: a right triangle

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

Example 3.2.4

One angle of a right triangle measures $28° .$ What is the measure of the third angle?

Answer

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.	$.$

Your Turn 3.2.4

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

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 3.2.5

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

Answer

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.	$.$

Your Turn 3.2.5

The measure of one angle of a right triangle is $50°$ more than the measure of the smallest angle. Find the measures of all three angles.

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 3.2.7 are similar. Each side of $Δ A B C$ is four times the length of the corresponding side of $Δ X Y Z$ and their corresponding angles have equal measures.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3. — Figure 3.2.7: $Δ A B C$ and $Δ X Y Z$ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

Properties of Similar Triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

$...$

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $Δ A B C$

The length $a$ can also be written $BC$
The length $b$ can also be written $AC$
The length $c$ can also be written $AB$

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

Example 3.2.6

$Δ A B C$ and $Δ X Y Z$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

$Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.$

Answer

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 $Δ A B C$ y = length of the third side $Δ X Y Z$
Step 4. Translate.	The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z}$ Since the side $A B = 4$ corresponds to the side $X Y = 3$ , we will use the ratio $\frac{AB}{XY} = \frac{4}{3}$ 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 $Δ A B C$ is 6 and the third side of $Δ X Y Z$ is 2.4.

Your Turn 3.2.6

$Δ A B C$ is similar to $Δ X Y Z$ . Find $a .$

$Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.$

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 $90°$ angle.

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 3.2.8: In a right triangle, the side opposite the $90°$ angle is called the hypotenuse and each of the other sides is called a leg.

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 $Δ A B C,$

$a^{2} + b^{2} = c^{2}$

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. As a review, $\sqrt{25}$ is $5$ because $5^{2} = 25 .$

Example 3.2.7

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

$Right triangle with legs labeled as 3 and 4.$

Answer

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 $c =$ 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.

Your Turn 3.2.7

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

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

Example 3.2.8

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

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

Answer

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 $b =$ 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.

Your Turn 3.2.8

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

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

Example 3.2.9

Kelvin is building a gazebo and wants to brace each corner by placing a 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.

$A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.$

Answer

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:	$.$
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Your Turn 3.2.9

John puts the base of a 13 foot ladder 5 feet from the wall of his house. How far up the wall does the ladder reach?

$A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.$

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Supplementary and Complementary Angles

How To

Use a Problem Solving Strategy for Geometry Applications.

Example 3.2.1

Your Turn 3.2.1

Example 3.2.2

Your Turn 3.2.2

Sum of the Measures of the Angles of a Triangle

Example 3.2.3

Your Turn 3.2.3

Right Triangles

Example 3.2.4

Your Turn 3.2.4

Example 3.2.5

Your Turn 3.2.5

Similar Triangles

Properties of Similar Triangles

Example 3.2.6

Your Turn 3.2.6

Use the Pythagorean Theorem

The Pythagorean Theorem

Example 3.2.7

Your Turn 3.2.7

Example 3.2.8

Your Turn 3.2.8

Example 3.2.9

Your Turn 3.2.9

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