Use Properties of Angles, Triangles, and the Pythagorean Theorem m81326 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 4c0177e5-a020-44df-8147-f6599ea62f38 Before you get started, take this readiness quiz. Solve: x+3+6=11. If you missed this problem, review . x=2 Solve: a45=43. If you missed this problem, review . 60 Simplify: 36+64. If you missed this problem, review . 10 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 AnglesAre 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. See .
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 , ∠A is the angle with vertex at point A. The measure of ∠A is written m∠A.
∠A is the angle with vertex at pointA.
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°, then they are called supplementary angles. In , each pair of angles is supplementary because their measures add to 180°. Each angle is the supplement of the other.
The sum of the measures of supplementary angles is 180°.
If the sum of the measures of two angles is 90°, then the angles are complementary angles. In , each pair of angles is complementary, because their measures add to 90°. Each angle is the complement of the other.
The sum of the measures of complementary angles is 90°.
Supplementary and Complementary AnglesIf 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 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. 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. An angle measures 40°. Find ⓐ its supplement, and ⓑ its complement. 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 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.
An angle measures 25°. Find its: ⓐ supplement ⓑ complement. ⓐ 155°ⓑ 65° An angle measures 77°. Find its: ⓐ supplement ⓑ complement. ⓐ 103°ⓑ 13° Did you notice that the words complementary and supplementary are in alphabetical order just like 90 and 180 are in numerical order? Two angles are supplementary. The larger angle is 30° more than the smaller angle. Find the measure of both angles. 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.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.
Two angles are supplementary. The larger angle is 100° more than the smaller angle. Find the measures of both angles. 40°, 140° Two angles are complementary. The larger angle is 40° more than the smaller angle. Find the measures of both angles. 25°, 65°
Use the Properties of TrianglesWhat do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in is called ΔABC, read ‘triangle ABC’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.
ΔABC has vertices A,B,andC and sides a,b,andc.
The three angles of a triangle are related in a special way. The sum of their measures is 180°. Sum of the Measures of the Angles of a Triangle For any ΔABC, the sum of the measures of the angles is 180°. The measures of two angles of a triangle are 55° and 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.
The measures of two angles of a triangle are 31° and 128°. Find the measure of the third angle. 21° A triangle has angles of 49° and 75°. Find the measure of the third angle. 56°
Right TrianglesSome triangles have special names. We will look first at the right triangle. A right triangle has one 90° angle, which is often marked with the symbol shown in .
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. One angle of a right triangle measures 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.
One angle of a right triangle measures 56°. What is the measure of the other angle? 34° One angle of a right triangle measures 45°. What is the measure of the other angle? 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. 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. 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.
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. 20°, 70°, 90° The measure of one angle of a right triangle is 30° more than the measure of the smallest angle. Find the measures of all three angles. 30°, 60°, 90°
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 are similar. Each side of ΔABC is four times the length of the corresponding side of ΔXYZ and their corresponding angles have equal measures.
ΔABC and ΔXYZ 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 ΔABC: the lengthacan also be writtenBCthe lengthbcan also be writtenACthe lengthccan also be writtenAB We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths. ΔABC and ΔXYZ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle. Solution 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. Leta = length of the third side of ΔABCy = length of the third side ΔXYZ Step 4. Translate. The triangles are similar, so the corresponding sides are in the same ratio. SoSince the side AB=4 corresponds to the side XY=3, we will use the ratio ABXY=43 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 ΔABC is 6 and the third side of ΔXYZ is 2.4.
ΔABC is similar to ΔXYZ. Find a. 8 ΔABC is similar to ΔXYZ. Find y. 22.5
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, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° angle is called the hypotenuse, and the other two sides are called the legs. See .
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 TheoremIn any right triangle ΔABC, where c is the length of the hypotenuse a and b are the lengths of the legs. 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 m and defined it in this way: For example, we found that 25 is 5 because 52=25. We will use this definition of square roots to solve for the length of a side in a right triangle. Use the Pythagorean Theorem to find the length of the hypotenuse. 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=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.
Use the Pythagorean Theorem to find the length of the hypotenuse. 10 Use the Pythagorean Theorem to find the length of the hypotenuse. 17 Use the Pythagorean Theorem to find the length of the longer leg. 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=the leg of the triangleLabel 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.
Use the Pythagorean Theorem to find the length of the leg. 8 Use the Pythagorean Theorem to find the length of the leg. 12 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. 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.
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? 12 feet Randy wants to attach a 17-ft string of lights to the top of the 15-ft mast of his sailboat. How far from the base of the mast should he attach the end of the light string? 8 feet ACCESS ADDITIONAL ONLINE RESOURCES Animation: The Sum of the Interior Angles of a Triangle Similar Polygons Example: Determine the Length of the Hypotenuse of a Right Triangle
Key Concepts Supplementary and Complementary AnglesIf 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. Solve Geometry ApplicationsRead 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. Sum of the Measures of the Angles of a Triangle For any ΔABC, the sum of the measures is 180° m∠A+m∠B+m∠C=180 Right Triangle A right triangle is a triangle that has one 90° angle, which is often marked with a ⦜symbol. Properties of Similar TrianglesIf two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.
Practice Makes PerfectUse the Properties of Angles In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle. 53° ⓐ 127°ⓑ 37° 16° 29° ⓐ 151°ⓑ 61° 72° In the following exercises, use the properties of angles to solve. Find the supplement of a 135° angle. 45° Find the complement of a 38° angle. Find the complement of a 27.5° angle. 62.5° Find the supplement of a 109.5° angle. Two angles are supplementary. The larger angle is 56° more than the smaller angle. Find the measures of both angles. 62°, 118° Two angles are supplementary. The smaller angle is 36° less than the larger angle. Find the measures of both angles. Two angles are complementary. The smaller angle is 34° less than the larger angle. Find the measures of both angles. 62°, 28° Two angles are complementary. The larger angle is 52° more than the smaller angle. Find the measures of both angles. Use the Properties of Triangles In the following exercises, solve using properties of triangles. The measures of two angles of a triangle are 26° and 98°. Find the measure of the third angle. 56° The measures of two angles of a triangle are 61° and 84°. Find the measure of the third angle. The measures of two angles of a triangle are 105° and 31°. Find the measure of the third angle. 44° The measures of two angles of a triangle are 47° and 72°. Find the measure of the third angle. One angle of a right triangle measures 33°. What is the measure of the other angle? 57° One angle of a right triangle measures 51°. What is the measure of the other angle? One angle of a right triangle measures 22.5°. What is the measure of the other angle? 67.5° One angle of a right triangle measures 36.5°. What is the measure of the other angle? The two smaller angles of a right triangle have equal measures. Find the measures of all three angles. 45°, 45°, 90° The measure of the smallest angle of a right triangle is 20° less than the measure of the other small angle. Find the measures of all three angles. 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. 30°, 60°, 90° The angles in a triangle are such that the measure of one angle is 20° more than 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. Find the Length of the Missing Side In the following exercises, ΔABC is similar to ΔXYZ. Find the length of the indicated side. side b 12 side x 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. Find the distance from Los Angeles to San Francisco. 351 miles Find the distance from San Francisco to Las Vegas. Use the Pythagorean Theorem In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse. 15 25 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. 8 12 10.2 16.2 In the following exercises, solve. Approximate to the nearest tenth, if necessary. 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 should the end of the string of lights be anchored? 5 feet Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is 12 feet high and 16 feet wide. How long should the banner be to fit the garage door? 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? 14.1 feet Brian borrowed a 20-foot extension ladder to paint his house. If he sets the base of the ladder 6 feet from the house, how far up will the top of the ladder reach?
Everyday Math 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? 2.9 feet Measurement A city engineer plans to build a footbridge across a lake from point X to point Y, as shown in the picture below. To find the length of the footbridge, she draws a right triangle XYZ, with right angle at X. She measures the distance from X to Z,800 feet, and from Y to Z,1,000 feet. How long will the bridge be?
Writing Exercises Write three of the properties of triangles from this section and then explain each in your own words. Answers will vary. Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length 3 and 4.
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?
angle An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle. complementary angles If the sum of the measures of two angles is 90°, then they are called complementary angles. hypotenuseThe side of the triangle opposite the 90° angle is called the hypotenuse. legs of a right triangle The sides of a right triangle adjacent to the right angle are called the legs. right triangle A right triangle is a triangle that has one 90° angle. similar figures In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. supplementary angles If the sum of the measures of two angles is 180°, then they are called supplementary angles. triangle A triangle is a geometric figure with three sides and three angles. vertex of an angle When two rays meet to form an angle, the common endpoint is called the vertex of the angle.