MATH STUDENT BOOK. 8th Grade Unit 6


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1 MATH STUDENT BOOK 8th Grade Unit 6
2 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular and Parallel Lines, Part 2 20 Circles 25 SELF TEST 1: Angle Measures and Circles Polygons 37 Classifying Polygons 37 Interior and Exterior Measures of Polygons 42 Classifying Triangles and the Triangle Inequality Theorem 50 The Quadrilateral Family 57 SELF TEST 2: Polygons Indirect Measure 65 Pythagorean Theorem, Part 1 65 Pythagorean Theorem, Part 2 70 SELF TEST 3: Indirect Measure Review 77 LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. Section 1 1
3 Measurement Unit 6 Author: Glynlyon Staff Editor: Alan Christopherson, M.S. Westover Studios Design Team: Phillip Pettet, Creative Lead Teresa Davis, DTP Lead Nick Castro Andi Graham Jerry Wingo 804 N. 2nd Ave. E. Rock Rapids, IA MMXIV by Alpha Omega Publications a division of Glynlyon, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, Inc. All trademarks and/or service marks referenced in this material are the property of their respective owners. Alpha Omega Publications, Inc. makes no claim of ownership to any trademarks and/ or service marks other than their own and their affiliates, and makes no claim of affiliation to any companies whose trademarks may be listed in this material, other than their own. Some clip art images used in this curriculum are from Corel Corporation, 1600 Carling Avenue, Ottawa, Ontario, Canada K1Z 8R7. These images are specifically for viewing purposes only, to enhance the presentation of this educational material. Any duplication, resyndication, or redistribution for any other purpose is strictly prohibited. Other images in this unit are 2009 JupiterImages Corporation 2 Section 1
4 Unit 6 Measurement Measurement Introduction This unit covers basic geometric concepts associated with angles, parallel and perpendicular lines, and circles. The properties of various polygons and their angle measures are used to solve for missing angle measures. The special relationship that exists between the sides and angles of right triangles, including a proof of the Pythagorean theorem, is also discussed. Objectives Read these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC. When you have finished this LIFEPAC, you should be able to: z Classify and measure angles and lines. z Identify and find measures of angles created by transversals. z Identify parts of circles and their measures. z Classify polygons and find measures of their interior and exterior angles. z Classify triangles and use the triangle inequality theorem. z Classify quadrilaterals and the relationships among them. z Find side lengths of right triangles using the Pythagorean theorem. Section 1 3
5 Measurement Unit 6 Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here. 4 Section 1
6 Unit 6 Measurement 1. Angle Measures and Circles Classify and Measure Angles Math is part of our everyday world in everything from balancing checkbooks, to shopping, to measuring a room for new carpet. Math is also a key part of many of the games we play. For instance, the game of pool is nothing but a game of angles. Knowing the correct angle needed to hit the ball is essential to winning a game of pool. One of the best skills you can have if you are a pool player is being able to identify the angle and angle measure needed to sink a shot. Objectives Identify angles by their measure. Classify pairs of angles. Find the measure of an angle. Vocabulary acute angle an angle that measures more than 0 but less than 90 adjacent angles two angles that have a common vertex and side but are not overlapping complementary angles two angles whose sum is 90 obtuse angle an angle that measures more than 90 but less than 180 right angle an angle that measures exactly 90 straight angle an angle that measures exactly 180 supplementary angles two angles whose sum is 180 vertex the point where two line segments, lines, or rays meet to form an angle vertical angles angles that are opposite from one another at the intersection of two lines; vertical angles are congruent Section 1 5
7 Measurement Unit 6 Classifying Angles Angles are a part of our everyday life. Just by looking around the room you are in, you should be able to see hundreds of angles! Each angle or pair of angles has a name and set of properties to define it. You may remember the terms acute angle, right angle, obtuse angle, and straight angle. An acute angle is an angle that measures more than 0 but less than 90. A right angle measures exactly 90, while an obtuse angle measures more than 90 but less than 180. A straight angle, or straight line, measures exactly 180. You can easily classify an angle as acute, right, obtuse, or straight just by looking at it. If you remember that a right angle is like a corner on a piece of paper and a straight angle is a straight line, you can tell what type of angle you have without measuring it. If the angle is smaller than the corner of a piece of paper, you have an acute angle. If it is larger than the corner of a piece of paper but smaller than a line, then you have an obtuse angle. Let s look at how to classify pairs of angles. Adjacent angles are two angles that share a common vertex, point, and side. They are next to one another without overlapping. Two types of angles that are often adjacent are complementary angles and supplementary angles. Complementary angles are two angles whose sum is 90. The two angles can either be adjacent angles or separate angles. The following graphic shows two adjacent, complementary angles. Complementary angles: Supplementary angles are two angles whose sum is 180 ; the angles can either be separate or adjacent. The following graphic shows two adjacent, supplementary angles. Supplementary angles: Angles are also created when lines intersect. When the two lines intersected, they created four angles. The four angles all share a common vertex, where the lines intersect. Some of the angles can also be described as being adjacent. Angle 1 is adjacent to both angle 4 and angle 2. Angle 2 is adjacent to angle 1 and angle 3. Angle 3 is adjacent to angles 2 and 4. Finally, angle 4 is adjacent to angles 3 and 1. 6 Section 1
8 Unit 6 Measurement Within these four angles, there are 2 sets of vertical angles. Vertical angles are angles that are opposite one another but share a vertex. Angles 1 and 3 are vertical angles, because they have the same vertex and are opposite one another. The other set of vertical angles are angles 2 and 4. Again, they have the same vertex and are opposite one another. Vertical angles are easily identifiable, because they are found where lines intersect. Look at the following picture. Three lines intersect at one point. The newly created angles are labeled for you. There are four sets of vertical angles: 1 and 3 2 and 4 5 and 7 6 and 8 Let s look at more sets of vertical angles. See if you can identify them before reading the answers. There are 6 sets of vertical angles: 1 and 4 2 and 3 5 and 8 6 and 7 9 and and 11 Measurements of Angles Now that you know how to classify angles and pairs of angles, we can look into how to find the exact measure of an angle. Remember that complementary angles have a sum of 90, and supplementary angles have a measure of 180. Look at the following examples to see how to find the measure of the angle. Section 1 7
9 Measurement Unit 6 x + 39 = 180 Supplementary angles sum to 180. Subtract 39 from x = both sides. Complete the x = 141 subtractions. The first thing you need to determine is if the pair of angles are complementary or supplementary. The angles are complementary. We can tell this because there is a right angle symbol. We now know that the two angles have a sum of 90. We also know that one of the angles measures x, while the other angle measures 72. We have enough information to set up an equation to find the value of x. x + 72 = 90 Complementary angles sum to 90. Subtract 72 from x = both sides. Complete the x = 18 subtractions. The second angle in the complementary pair of angles measures 18. This time we have a pair of supplementary angles, so we need to set the equation equal to 180. Let s move on to vertical angles. You know that vertical angles are angles that are opposite one another and share a vertex. Vertical angles are also congruent, or equal to one another. This means once you know the measure of one, you automatically know the measure of the other. 1 = 82 What are the measures of the other angles? 1 is a vertical angle with 3. Therefore, we now know that 3 equals 82. We can easily find angles 2 and 4. We know that they are also a set of vertical angles. This means that all we have to do is find the measure of one angle, and we ll know the measure of the other. If we look closely at the image, we see that angle 1 and angle 2 are supplementary, because they form a straight angle. Since supplementary angles equal 180, angle 1 equals 82, and angle 2 equals x, you can now set up an equation. 8 Section 1
10 Unit 6 Measurement x + 82 = 180 x = x = 98 Supplementary angles sum to 180. Subtract 82 from both sides. Complete the subtractions. x = 180 x = x = 45 Supplementary angles sum to 180. Subtract 135 from both sides. Complete the subtractions. Angle 2 measures 98. That means angle 4 also equals = = = 65 What are the measures of the other angles? The easiest way to solve this problem is to use one point of intersection at a time. Let s begin with the intersection that creates angles 1, 2, 3, and 4. So far, we know that 2 = 135. We also know that angles 2 and 3 are vertical angles, which means that they are equal. Finally, we can see that angles 1 and 2 are supplementary. That means they have a sum of 180. We now know that angle 1 equals 45. Since angle 1 and angle 4 are vertical angles, we can determine that angle 4 is also equal to 45. Let s move to the intersection that creates angles 5, 6, 7, and 8. You were told that 8 = 160. That means angle 5 also equals 160, because 8 and 5 are vertical angles. You can see that angles 5 and 6 are supplementary angles, so you know their sum is equal to 180. x = 180 x = x = 20 Supplementary angles sum to 180. Subtract 160 from both sides. Complete the subtractions. If angle 6 equals 20, then angle 7 also equals 20. Finally, let s look at the intersection that creates angles 9, 10, 11, and 12. You know that angle 11 equals 65, so that means angle 9 also equals 65. Angles 9 and 10 are supplementary to one another. x + 65 = 180 Supplementary angles sum to 180. Subtract 65 from x = both sides. Complete the x = 115 subtractions. Angle 10 measures 115, as well as angle 12. Section 1 9
11 Measurement Unit 6 Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. Identify the types of angles. Classify pairs of angles. Find the measure of angles. Complete the following activities. 1.1 If angle 1 measures 98 and angle 2 measures 72, then the two angles are supplementary. True False 1.2 Complementary angles have a sum of 90. True False 1.3 All pairs of vertical angles are equal. True False 1.4 Adjacent angles are always supplementary. True False 1.5 If angle 1 and angle 5 are vertical angles and angle 1 equals 55, then angle 5 will equal can t be determined 1.6 If each of two complementary angles has the same measure, then each angle will equal Section 1
12 Unit 6 Measurement 1.7 In the following diagram, angle 7 equals 61. What is the measure of angle 8? If two angles are supplementary and one of the angles measures 38, what is the measure of the larger angle? Use this illustration for the following questions. 1.9 Angle 3 is equal to angle Angle 2 is equal to angle Angle 7 is equal to angle Angle 6 is equal to angle. Section 1 11
13 Measurement Unit 6 Perpendicular and Parallel Lines, Part 1 Pictured is the Hearst Tower in New York. The tower earned the 2006 Emporis Skyscraper Award for the best skyscraper in the world completed that year. It was also the first green building in New York. Take a closer look at the triangular framing design on the outside of the building. Can you identify the different types of angles that are created? Can you also identify the different types of lines that exist? Objectives Identify lines as parallel, intersecting, or perpendicular. Identify a transversal and the angles it creates. Find the measure of angles created by a transversal. Vocabulary alternate exterior angles two outside angles that lie on different sides of a transversal alternate interior angles two inside angles that lie on different sides of a transversal corresponding angles two angles in the same position on different lines exterior angles outside angles interior angles inside angles intersecting lines lines that share one point parallel lines lines that never cross one another and are the same distance apart at all times perpendicular lines lines that intersect and create right angles transversals lines that intersect two or more lines to create angles Classifying Lines Every set of lines can be classified as parallel, perpendicular, or intersecting. They are often easy to tell apart, just by looking at them. distance apart. Intersecting lines are lines that cross one another at one point. Finally, perpendicular lines are lines that cross one another and create four right angles at the point of intersection. Parallel lines are two lines that never intersect. They are also always the same 12 Section 1
14 Unit 6 Measurement Sometimes, another line will cross a set of lines in a different direction. This individual line is called a transversal. A transversal is a line that crosses two or more lines and creates angles. Take a look at some examples of transversals. The transversals are the red lines in each picture. Notice that the transversal can cut horizontally, vertically, or diagonally. The transversal also creates two sets of angles known as alternate exterior angles. Alternate exterior angles are angles that are on the opposite sides of the transversal, opposite line, and on the outside of the lines. The last two sets of angles created by the transversal are alternate interior angles. Alternate interior angles are angles that are on the opposite sides of the transversal, opposite line, and are on the inside of the lines. When a transversal cuts across a set of two lines, it creates eight angles, four angles per intersection. The angles created on the two lines have a relationship with one another. One type of angle that is created is called corresponding angles. Corresponding angles are angles that are in the same position but on different lines. There are four sets of corresponding angles. When a transversal cuts across a set of parallel lines, the angles that are created have a special relationship. Corresponding angles, alternate interior angles, and alternate exterior angles are all congruent within their own pairs. Key point! The sets of angles only become congruent when a transversal cuts across a set of parallel lines! Section 1 13
15 Measurement Unit 6 Identify the two sets of alternate interior angles. Assume the lines are parallel. 2 and 7 3 and 6 Identify the two sets of alternate exterior angles. Assume the lines are parallel. 1 and 3 2 and 4 5 and 7 6 and 8 Let s look at some examples where we can use congruency to find the measures of the eight angles created by a transversal. What are the measures of each angle? 1 and 7 2 and 8 Identify the four sets of corresponding angles. Assume the lines are parallel. line a line b 2 = 70 Let s begin by finding the value of 3. You know that angles 2 are 3 are vertical angles, and that vertical angles are congruent. So, the measure of angle 3 is Section 1
16 Unit 6 Measurement You also know that angle 1 and 2 are supplementary, which means their sum is equal to 180. Since angle 2 measures 70, then angle 1 must measure 110. You can now use either vertical angles or supplementary angles to find the measure of angle 4. Either way, angle 4 will measure 110. Take a look at what you have so far. Finally, use vertical angles to find angle 7. Since angles 6 and 7 are vertical, and angle 6 measures 70, then angle 7 also measures 70. Here is the completed diagram with angle measures. Now, let s use the new angle relationships you just learned about. You can use corresponding angles, alternate interior angles, or alternate exterior angles to move from line a to line b. Let s use corresponding angles this time. Angle 1 is in the upper left of the intersection on line a. Angle 5 is in the upper left of the intersection on line b. Because lines a and b are parallel, you know that angle 1 and angle 5 are congruent. So, angle 5 has a measure of 110. Now you can use vertical angles to find that the measure of angle 8 is also 110. Again, use supplementary angles to find the measure of angle 6. Since angle 5 measures 110, then angle 6 must measure 70. See if you can spot the four sets of corresponding angles, the two sets of alternate interior angles, and the two sets of alternate exterior angles. Is each set congruent? Find the measure of each angle. line x line y 6 = 47 You know that angle 1 is a vertical angle to angle 6; so, it also must have a measure of 47. You also know that angle 1 and angle 2 are supplementary, so they have Section 1 15
17 Measurement Unit 6 a sum of 180. Since 47 of the 180 is occupied by angle 1, that leaves angle 2 to measure 133. Angle 5 is a vertical angle to angle 2; so, that means it also has a measure of 133. This time, let s use alternate exterior angles to move between the lines. Angle 5 is an outside angle, so its alternate exterior angle is angle 4. Since the lines are parallel, these two angles are congruent, giving angle 4 a measure of 133. Angle 4 is a vertical angle to angle 7; so, angle 7 also has a measure of 133. That leaves angles 3 and 8 to find. Angle 3 corresponds to angle 1; so that means they have the same measure of 47. Since angle 3 and angle 8 are vertical, angle 8 also has a measure of 47. There is no one correct way to find the measures of the angles, so don t worry if you and your friend have different ways of finding the answers. Let s see what happens when the set of lines being crossed by the transversal aren t parallel. Find the measures of each angle, if m 1 = 61 and m 6 = 102. Be Careful! Remember that transversals always create sets of corresponding angles, alternate interior angles, and alternate exterior angles. However, the sets of angles are only congruent when the transverse set of lines is parallel. Let s start with line m. Angles 1 and 4 are vertical angles; so, angle 4 must also measure 61. Angles 1 and 2 are supplementary angles; so, that means that angle 2 measures 119. Finally, for line m, angles 2 and 3 are vertical angles; so, angle 3 must also measure 119. Even though corresponding angles, alternate exterior angles, and alternate interior angles exist between the two lines, they are not congruent, so we can t equate the measures. If you look back at the original problem, you were given the measure for angle 6. Since angle 6 is a vertical angle with angle 7, you know that the measure of angle 7 is also 102. Angle 6 and angle 8 are supplementary angles; so, that means angle 8 has to have a measure of 78. Angle 8 is a vertical angle with angle 5, which means that angle 5 also has a measure of Section 1
18 Unit 6 Measurement Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. Identify lines as parallel, intersecting, or perpendicular. Identify a transversal and the angles it creates. Identify the sets of angles created by a transversal. Find the measure of angles created by a transversal. Complete the following activities. Use the following illustration for Assume the lines are parallel Select all pairs of corresponding angles. 1 and 5 2 and 6 2 and 5 3 and 7 3 and Select all pairs of alternate interior angles. 3 and 4 3 and 6 3 and 5 4 and Select all pairs of alternate exterior angles. 1 and 2 1 and 8 1 and 7 2 and 7 4 and 7 4 and 8 4 and 6 5 and 6 2 and 8 7 and 8 1 and 6 none none none Section 1 17
19 Measurement Unit 6 Use the following illustration for Assume the lines are parallel Select all of the angles that have the same measure as angle Select all of the angles that have the same measure as angle can t be determined 8 can t be determined 1.18 The value of x that makes lines a and b parallel is. 18 Section 1
20 Unit 6 Measurement 1.19 The measure of angle 7 is Line is a transversal. Use the following illustration for Assume the lines are parallel Angle 4 corresponds with angle Find the measure of each angle. m 2 =. m 3 =. m 4 =. m 6 =. m 7 =. m 8 =. m 5 =. Section 1 19
21 Measurement Unit 6 Perpendicular and Parallel Lines, Part 2 What is the only angle from which you should deal with a problem? Answer: Q ZKNQFUST! Can you figure out what the answer to the question is? Try using the secret decoder ring to help figure it out. Use the outside circle for the letter in the code. Replace the code letter with the letter on the inner circle. This lesson focuses on finding the measures of angles when there are variables in the angles. It is similar to being written in code, except this code is in algebraic form with variables. Objectives Identify the relationships between angles created by a transversal across parallel lines. Find the measure of the angles created by a transversal across parallel lines. Finding Angle Measures The only angle from which you should deal with a problem is a TRYangle! The study of writing in code is called cryptography. Sometimes governments send coded messages, and, sometimes, you and your friends do, too. Have you ever written a note in code, so that only you and the person you are writing to know what it says? Previously, you learned about vertical angles, complementary angles, supplementary angles, and angles created by transversals. We are going to continue our study of these angles, but this time we are going to have variables and variable expressions as our angle measures. Just like any other time you see a variable in an equation, you need to find the value of the variable to find the angle measure. Let s begin by working with complementary and supplementary angles. Remember that complementary angles have a sum of 90, and supplementary angles have a sum of Section 1
22 Unit 6 Measurement Find the measure of the two angles. We have a pair of supplementary angles. So, we need to set the equation equal to 180. Be careful the value of 15 doesn t tell us either angle measure. It does, however, help us, because we now know the value of x. All we have to do is plug 15 into the expression for each angle. x + 3x = 180 4x = 180 = Supplementary angles sum to 180. Complete the addition on the left. Divide both sides by 4. 4x = 4(15 ) = 60 2x = 2(15 ) = 30 x = 45 Complete the division. We know that x is 45. That means we can also find the other angle of 3x. 3x = 3(45 ) = 135 The two angle measures are 45 and 135. We have a set of supplementary angles, which sum to 180. Find the measure of each angle. This is a set of complementary angles. So, the sum of the angles will be equal to 90. 4x + 2x = 90 6x = 90 = Complementary angles sum to 90. Complete the addition on the left. Divide both sides by 6. x = 15 Complete the division. 2x + (5x + 33 ) = 180 Supplementary angles sum to x + 33 = 180 Add the variables. Subtract 33 from 7x = both sides. = x = 21 Divide both sides by 7. Complete the division. So far, we know that x = 21. But, that doesn t tell us either angle measure. To find the angle measures, we need to plug 21 in for x in each expression. 2x = 2(21 ) = 42 5x + 33 = 5(21 ) + 33 = = 138 Section 1 21
23 Measurement Unit 6 The two angles measure 42 and 138. You can check your work by adding them together to make sure you get a sum of 180. x + 34 = = 51 Now, let s look at what happens when we replace angle measures in vertical angles. Remember that vertical angles are always equal to one another. Find the measure of the shaded angle. Find the value of x and, then, the measure of the vertical angles. Since vertical angles are equal in measure, we need to put each angle on either side of the equal sign. 3x = x + 34 You solve this equation just as you would solve any other equation, by solving for x. 3x = x + 34 Vertical angles are congruent. 3x  x = x  x + 34 Subtract x from both sides. 2x = 34 Simplify. = Divide both sides by 2. x = 17 Complete the division. Now that you know the value of x, plug that value into each angle s expression to make sure that they are equal to one another. 3x = 3(17 ) = 51 We see that the two angles for which we have values are supplementary angles. So, we can add them together to get a sum of 180. (4x + 25 ) + (7x  21 ) = x + 4 = x = x = 176 = x = 16 Supplementary angles sum to 180. Combine the variables and constants on the left side. Subtract 4 from both sides. Complete the subtraction. Divide both sides by 11. Complete the division. Now, plug that value into the expression 7x  21, because it is a vertical angle to the shaded angle. 7x  21 = 7(16 )  21 = = 91 The measure of the shaded angle is Section 1
24 Unit 6 Measurement Finally, let s take a look at angles that are created by a transversal. Find the measure of each angle. You are given that m 5 = 3x and m 6 = 2x Angles 5 and 6 are supplementary angles; so, we know they have a sum of 180. Therefore, we begin by adding their values and setting it equal to x + (2x + 25 ) = 180 Supplementary angles sum to 180. Combine the 5x + 25 = 180 variables on the left side. Subtract 25 from 5x = both sides. Complete the 5x = 155 subtraction. = x = 31 Divide both sides by 5. Complete the division. This just tells us the value of x, not an angle measure. We need to plug 31 into each of the expressions to get the angle measures. m 5 = 3x = 3(31 ) = 93 Angle 5 measures 93. m 6 = 2x + 25 = 2(31 ) + 25 = = 87 Angle 6 measures 87. Now you can use vertical angles to finish this intersection. Angle 2 is a vertical angle with angle 5; so, it measures 93. Angle 1 is a vertical angle with angle 6; so, it has a measure of 87. If you use corresponding angles, angle 7 corresponds with angle 5; so, angle 7 also measures 93. If angle 7 measures 93, then angle 4 also measures 93. Angle 8 corresponds to angle 6; so, it has a measure of 87. Because angle 3 is a vertical angle with angle 8, it too has a measure of 87. Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. A transversal across parallel lines creates corresponding, alternate interior, and alternate exterior angles. You can use congruence among angles, complementary relationships, and supplementary relationships to calculate the measures of unknown angles. Section 1 23
25 Measurement Unit 6 Complete the following activities Use the following diagram to answer the questions. Given A B. The value of x is. The measure of 5 is. The measure of 1 is. The measure of 6 is. The measure of 2 is. The measure of 7 is. The measure of 3 is. The measure of 8 is. The measure of 4 is. Angles 2 and 3 are angles Use the diagram and information to match the following angles with their correct measures. Given Z = 2b + 6 and Y = 3b  1. W X Y Z Section 1
26 Unit 6 Measurement Circles Think back to kindergarten when you first began to learn about shapes. You knew that the triangle had three sides, the square had four sides, and that the circle was the round shape without any sides. You could even identify things around the classroom that were those shapes. But, do you know that there is more to these shapes, especially the circle, then what they taught you in kindergarten? This lesson will focus on the different parts of a circle. Objectives Identify the parts of a circle. Classify angles of circles. Find the measures of arcs and angles of circles. Vocabulary arc part of the circumference of a circle between two points central angle an angle that has the center of a circle as its vertex chord a line segment that connects two points on a circle circle a figure with all of its points the same distance from the center circumference of a circle the length around the exterior of a circle; measured in degrees in terms of an arc diameter a line segment that goes through the center of a circle to connect two points on the circle inscribed angle an angle in a circle with a vertex on the circle and sides that are chords of the circle intercepted arc the part of the circumference of a circle that is bound by two endpoints of a chord, radius, secant line or tangent line major arc an arc with a measure greater than 180 degrees minor arc an arc with a measure less than 180 degrees radius a line segment from the center of a circle to any point on the circle secant a line that goes through the circle at two distinct points on the circumference of a circle; the distance between the two points is considered a chord Section 1 25
27 Measurement Unit 6 semicircle an arc with a measure of 180 degrees tangent a line that touches the exterior part of the circumference of a circle at exactly one point You know that a circle is a round shape that doesn t have any sides, but do you know its real definition? A circle is a figure that has all of its points the same distance from the center. A circle is named by its center point. Look at the following figure. Notice the center dot labeled A. Each of the points that surround this point is the same distance as the other points. Now that you know what a circle really is, let s look into the parts of the circle. Sometimes, there are line segments that connect two points on the circle. This line segment might not always go through the center of the circle. If a line segment connects two points on the circle, then it is called a chord. Look at the picture of some examples of chords. Chords do not all have the same measure because they can have different lengths. A chord can sometimes be a diameter but a chord can never be a radius. Inside the Circle The radius is a line segment that connects the center of a circle to any point on the circle. It doesn t matter where you connect the center of the circle to the edge to form the radius, because all the points that make up the circle are the same distance from the center. Therefore, every line segment drawn from the center of the circle to the edge of the circle has the same measure for that circle. A diameter is a special type of chord. Like all other chords it connects two points on a circle. But, for a chord to be a diameter, it must go through the center of the circle. Another way to think of a diameter is two 26 Section 1
28 Unit 6 Measurement radiuses put together to form a straight line. The measure of a diameter is twice that of the radius. travels through the inside of a circle, while the secant does. Did you know! A radius is half the measure of the diameter. So, r = d and d = 2r. Angles of a Circle Outside the Circle So far, we have looked at line segments inside the circle, but what happens when there are lines, too? The first line we will look at is called a secant. A secant is made up of a chord but it continues past the edge of the circle. You know that an angle is two rays or line segments that share a common vertex. The same is true for angles found in a circle. The first kind of angle we are going to look at is called a central angle. A central angle is an angle that has the center point of the circle as its vertex. Each leg of a central angle is a radius of the circle. Look at the example where the central angle is BAD. Central Angle There is another line that is found completely on the outside of the circle. A tangent is a line that touches the circle at exactly one point. It is easy to tell a tangent from a secant, because a tangent line never There is another type of angle found within a circle. It is called an inscribed angle. An inscribed angle is an angle in a circle with a vertex on the circle and with sides that are chords of the circle. This is a little different from the central angle, because Section 1 27
29 Measurement Unit 6 an inscribed angle s vertex is not the center of the circle but a point on the circle. Look closely at the following picture; do you see the difference between the inscribed angle, XYZ, and the central angle shown previously? Inscribed Angle Reminder! A circle has a measure of 360. Arcs can be all different sizes. Similar to angles of triangles, we can classify arcs by their measure. A minor arc is an arc that has a measure less than 180. Arcs of a Circle What is an arc? An arc is part of a circle between two points on the circle. Look at the following examples. This might help! If an arc is half of the circle, then it is a semicircle. If the arc is less than half of the circle, then it is a minor arc. If the arc is more than half of the circle, then it is a major arc. A semicircle is an arc with a measure of exactly 180. Circle A has the following arcs: 28 Section 1
30 Unit 6 Measurement An arc that has a measure greater than 180 is called a major arc. There is also a special arc called an intercepted arc. An intercepted arc is an arc that is created by an inscribed angle. Remember that an inscribed angle is an angle whose vertex is on the circle. Thus, the intercepted arc is the part of the circle between the legs of this angle. Look at the following picture to help you visualize an intercepted arc. intercepted arc created by the angle. For example, if the measure of a central angle is 79, then the measure of the arc created is also 79. It also works the other way. If you know the measure of the arc created by a central angle is 17, then you know that the measure of the central angle is also 17. An inscribed angle is an angle whose vertex is on the circle and whose legs are chords of the circle. The intercepted arc created by the inscribed angle is two times larger than the measure of the inscribed angle. Let s look at some examples. What is the measure of arc BD? Phew! That was a lot of information to digest all at once. Before learning more about the measures of the angles and arcs of a circle, consider reviewing the definition and picture for each part of a circle. The first step is to determine if the angle that creates the arc is a central angle or an inscribed angle. The angle, BAD, is a central angle. That means the arc will have the same measure. The angle measures 63 ; so, arc BD also measures 63. Measures of Angles and Arcs Central angles are the angles whose vertex is the center of the circle and the legs of the angle are radii. The measure of a central angle is equal to the measure of the Section 1 29
31 Measurement Unit 6 What is the measure of arc HT? The arc is formed from an inscribed angle. So, this time the measure of the angle is half of the arc. To find the measure of the arc, multiply the measure of the angle by (2) = 148 What is the value of x? Since the arc is created by a central angle, we can set the two measures equal to one another. 5x + 9 = 144 5x = 135 x = 27 What is the measure of arc RB? This time, you know the angle has a measure of 62, and that it is an inscribed angle. This means that the arc measure is twice the angle s measure. 62 (2) = 124 What is the measure of x? We have an intercepted arc and an inscribed angle. That means that we need to multiply the angle measure by 2 and, then, set it equal to the arc measure. 30 Section 1
32 Unit 6 Measurement 2(3x + 10 ) = 104 6x + 20 = 104 6x = 84 x = 14 Be sure that you understand the relationships that exist between the different angles and the arcs that are created by them. A central angle and its arc have the same measure, while an inscribed angle has half the measure of the intercepted arc. Let s Review Before going on to the practice problems, make sure you understand the main points of this lesson. The parts of a circle include the center, equidistant points, central angles, inscribed angles, arcs, intercepted arcs and chords. The measure of a central angle is equal to the measure of its intercepted arc. The measure of an inscribed angle is half the measure of its intercepted arc. The measure of an intercepted arc is twice the measure of its inscribed angle. Complete the following activities A radius is the diameter A semicircle measures degrees A angle has the same measure as its arc A circle is named by its A diameter is also a If an inscribed angle measures 67, how would you find the intercepted arc? Set the angle and arc measures Multiply the angle measure by equal. two. Divide the angle measure by two A line that intersects a circle at exactly one point is called a. chord tangent secant Section 1 31
33 Measurement Unit If an arc measures 179, what kind of arc is it? minor arc major arc semicircle 1.33 If the measure of a central angle is 28, then to find the measure of its arc, you would need to do which of the following? Nothing, they are equal. Divide the angle measure by 2. Multiply the angle measure by A chord is a diameter. always sometimes never 1.35 If the diameter of the circle is 11 inches, then the radius will be inches If the measure of an intercepted arc is 118, and the measure of the inscribed angle is 2x  1, what is the equation to find the value of x? 2x  1 = 118 2x  1 = 2(118 ) 2(2x  1) = The measure of the arc created by a central angle is 57. What type of arc is this? minor arc major arc semicircle 1.38 The measure of an inscribed angle is 110. What is the measure of the intercepted arc? A secant line contains a chord. always sometimes never Review the material in this section in preparation for the Self Test. The Self Test will check your mastery of this particular section. The items missed on this Self Test will indicate specific area where restudy is needed for mastery. 32 Section 1
34 Unit 6 Measurement SELF TEST 1: Angle Measures and Circles Complete the following activities (6 points, each numbered activity) Match each picture with its correct label. complementary angles acute angle straight angle intersecting lines minor arc inscribed angle secant chord Section 1 33
35 Measurement Unit If a central angle measures 87, then its arc will measure. half as much the same twice as much 1.03 Given 1 = 123 and 8 = 7z What is the value of z, if lines a and b are parallel? What is the measure of x in the diagram? If an intercepted arc measures 124, what is the measure of its inscribed angle? the same, 124 twice the measure, 248 half the measure, Two angles are supplementary. One angle measures 2x, and the other angle measures 3x Which equation can you use to find the measure of each angle? 2x + 3x  15 = 90 2x  3x  15 = 90 2x  3x  15 = 180 2x + 3x  15 = If corresponding angles are on parallel lines, then their measure is the same. always sometimes never 1.08 The measure of a central angle is 3x + 18, and the measure of its arc is 147. Which equation can you use to find the value of x? 2(3x + 18) = 147 3x + 18 = 2(147 ) 3x + 18 = Section 1
36 Unit 6 Measurement 1.09 If two angles are complementary, then the sum of their angles is equal to. each other What is the measure of x? If one angle of a set of alternate interior angles on parallel lines measures 77, then the other angle also equals 77. always sometimes never In the graphic, one pair of vertical angles is In the image, the corresponding angle to angle 1 is angle. Section 1 35
37 Measurement Unit If one angle of a set of supplementary angles measures 77, then the other angle measures If one angle of a set of vertical angles measures 63, the sum of the vertical angles is SCORE TEACHER initials date 36 Section 1
38 MAT0806 May 14 Printing ISBN N. 2nd Ave. E. Rock Rapids, IA
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