And Now From a New Angle Special Angles and Postulates LEARNING GOALS

And Now From a New Angle Special Angles and Postulates LEARNING GOALS KEY TERMS. In this lesson, you will: Calculate the complement and supplement of an angle. Classify adjacent angles, linear pairs, and vertical angles. Differentiate between postulates and theorems. Differentiate between Euclidean and non-euclidean geometries. supplementary angles complementary angles adjacent angles linear pair vertical angles postulate theorem Euclidean geometry Linear Pair Postulate Segment Addition Postulate Angle Addition Postulate compliment is an expression of praise, admiration, or congratulations. Often A when someone does something noteworthy, you may pay them a compliment to recognize the person s accomplishments. Even though they are spelled similarly, the word complement means something very different. To complement something means to complete or to make whole. This phrase is used in mathematics, linguistics, music, and art. For example, complementary angles have measures that sum to 180 degrees making the straight angle whole. In music, a complement is an interval that when added to another spans an octave makes it whole. The film Jerry McGuire features the famous line You complete me, meaning that the other person complements them or that together they form a whole. So, a complement can be quite a compliment indeed! 171. Special Angles and Postulates 171

Problem 1 Supplementary and complementary angles are defined. Students use a compass to draw angles of specified measures. Then they use algebra to calculate the complement and supplement of different angles. PROBLEM 1 Supplements and Complements Two angles are supplementary angles if the sum of their angle measures is equal to 180º. 1. Use a protractor to draw a pair of supplementary angles that share a common side, and then measure each angle. Supplementary angles that share a side form a straight line, or a straight angle. Angle pairs appear frequently throughout geometry. Have students go through Problems 1 and, and make a table organizing the different types of angle pairs. The table should include a diagram to associate the terms with a visual image. Students should understand that some angle pairs hold properties of other angle pairs. For example, a linear pair contains both supplementary and adjacent angles. Grouping Have students complete Questions 1 through 7 with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Questions 1 through 7 What are the measures of your angles in the supplementary pair? How did you calculate the measure of the supplement of the angle? 15 55. Use a protractor to draw a pair of supplementary angles that do not share a common side, and then measure each angle. 70 110 3. Calculate the measure of an angle that is supplementary to /KJL. J K What are the measures of your angles in the complementary pair? How did you calculate the measure of the complement of the 6 angle? Is it possible for two angles to be both congruent and supplementary? Is it possible for two angles to be both congruent and complementary? How do you know when to set the sum of the measures of the two angles equal to 90? How do you know when to set the sum of the measures of the two angles equal to 180? L An angle that is supplementary to /KJL has a measure of 158º. 17 Chapter Introduction to Proof

Two angles are complementary angles if the sum of their angle measures is equal to 90º. 4. Use a protractor to draw a pair of complementary angles that share a common side, and then measure each angle. Complementary angles that share a side form a right angle. 57 33 5. Use a protractor to draw a pair of complementary angles that do not share a common side, and then measure each angle. 5 65 6. Calculate the measure of an angle that is complementary to /J. J 6 An angle that is complementary to /J has a measure of 8º.. Special Angles and Postulates 173

7. Determine the measure of each angle. Show your work and explain your reasoning. a. Two angles are congruent and supplementary. x 1 x 5 180 x 5 180 x 5 90 The measure of each angle is 90º. The sum of the angle measures is 180º. b. Two angles are congruent and complementary. x 1 x 5 90 x 5 90 x 5 45 The measure of each angle is 45º. The sum of the angle measures is 90º. c. The complement of an angle is twice the measure of the angle. x 1 x 5 90 3x 5 90 x 5 30 The measure of the angle is 30º. The measure of the complement of the angle is 3 30º, or 60º. The sum of the angle measures is 90º. d. The supplement of an angle is half the measure of the angle. x 1 0.5x 5 180 1.5x 5 180 x 5 10 The measure of the angle is 10º. The measure of the supplement of the angle is 1 (10º), or 60º. The sum of the angle measures is 180º. 174 Chapter Introduction to Proof

Grouping Have students complete Question 8 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Question 8 What do parts (a) and (b) have in common? What is different parts (a) and (b)? Share a general description for the solution methods of parts (a) and (b). 8. Determine the angle measures in each diagram. a. (x 14) (x 0) (x 14) 1 (x 1 0) 5 90 3x 1 6 5 90 3x 5 84 x 5 8 Measure of 1st angle 5 x 14 5 8 14 5 14 Measure of nd angle 5 x 1 0 5 (8) 1 0 5 56 1 0 5 76 The measures of the two angles are 14º and 76º. b. (x 30) (3x 10) (3x 1 10) 1 (x 30) 5 180 4x 0 5 180 4x 5 00 x 5 50 Measure of 1st angle 5 x 30 5 50 30 5 0 Measure of nd angle 5 3x 1 10 5 3(50) 1 10 5 150 1 10 5 160 The measures of the two angles are 0º and 160º.. Special Angles and Postulates 175

Problem Students explore adjacent angles, linear pairs, and vertical angles. Grouping Have students complete Question 1 with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Question 1 Do adjacent angles share a common vertex? Do adjacent angles share a common side? Is there more than one way to correctly draw angle so that it is adjacent to angle 1? Explain. PROBLEM Angle Relationships You have learned that angles can be supplementary or complementary. Let s explore other angle relationships. /1 and / are adjacent angles. /5 and /6 are not adjacent angles. 1 5 6 /3 and /4 are adjacent angles. /7 and /8 are not adjacent angles. 3 4 1. Analyze the worked example. Then answer each question. a. Describe adjacent angles. Two angles are adjacent if they share a common vertex and share a common side. 7 8 176 Chapter Introduction to Proof

b. Draw / so that it is adjacent to /1. 1 c. Is it possible to draw two angles that share a common vertex but do not share a common side? If so, draw an example. If not, explain why not. 1 d. Is it possible to draw two angles that share a common side, but do not share a common vertex? If so, draw an example. If not, explain why not. It is not possible. If two angles share a side, then they must be adjacent on the same line, and therefore share a common vertex. Adjacent angles are two angles that share a common vertex and share a common side.. Special Angles and Postulates 177

Grouping Have students complete Question with a partner. Then share their responses as a class. /1 and / form a linear pair. /5 and /6 do not form a linear pair. Guiding Questions for Share Phase, Question Does a linear pair of angles always form a line? Does a line have a degree measure? What is the sum of the measures of the two angles that form a linear pair? Are the two angles forming a linear pair of angles always adjacent angles? Are the two non-common sides of a linear pair always opposite rays? 1 5 6 /3 and /4 form a linear pair. /7 and /8 do not form a linear pair. 3 4. Analyze the worked example. Then answer each question. a. Describe a linear pair of angles. Two angles are a linear pair if they are adjacent and their noncommon sides form a line. 8 7 178 Chapter Introduction to Proof

b. Draw / so that it forms a linear pair with /1. 1 So, are the angles in a linear pair always supplementary? c. Name all linear pairs in the figure shown. 3 1 4 The linear pairs are /1 and /4, /1 and /3, / and /3, / and /4. d. If the angles that form a linear pair are congruent, what can you conclude? If the angles that form a linear pair are congruent, then the intersecting lines, line segments, or rays forming the linear pair are perpendicular. A linear pair of angles are two adjacent angles that have noncommon sides that form a line.. Special Angles and Postulates 179

Guiding Questions for Share Phase, Question 3 Are vertical angles adjacent or non-adjacent angles? Do intersecting lines always form vertical angles? Two intersecting lines form how many different pairs of vertical angles? Three intersecting lines form how many different pairs of vertical angles? Why do you suppose vertical angles always congruent? /1 and / are vertical angles. /5 and /6 are not vertical angles. 1 6 5 /3 and /4 are vertical angles. /7 and /8 are not vertical angles. 3 8 7 4 3. Analyze the worked example. Then answer each question. a. Describe vertical angles. Vertical angles are nonadjacent angles formed by two intersecting lines. 180 Chapter Introduction to Proof

b. Draw / so that it forms a vertical angle with /1. 1 c. Name all vertical angle pairs in the diagram shown. 3 1 4 /1 and / are vertical angles. /3 and /4 are also vertical angles. d. Measure each angle in part (c). What do you notice? Vertical angles are always congruent. Vertical angles are two nonadjacent angles that are formed by two intersecting lines, line segments, or rays.. Special Angles and Postulates 181

Grouping Have students complete Questions 4 and 5 with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Questions 4 and 5 Is there more than one way to draw a correct diagram to represent the conditional statements in Question 5? Identify the hypothesis of the conditional statements in Question 5. Identify the conclusion of the conditional statements in Question 5. 4. Determine m/aed. Explain how you determined the angle measure. A E D (x 94) (4x 5) C B Angles DEB and AEC are vertical angles, so they are congruent. (x 1 94) 5 (4x 1 5) 94 5 3x 1 5 69 5 3x 3 5 x m/aec 5 3 1 94 5 117 m/deb 5 117 Angles AEC and AED are supplementary. m/aec 1 m/aed 5 180 117 1 m/aed 5 180 m/aed 5 63 So, the measure of angle AED is 63. 5. For each conditional statement, draw a diagram and then write the hypothesis as the Given and the conclusion as the Prove. a. m/deg 1 m/gef 5 180º, if /DEG and /GEF are a linear pair. Given: /DEG and /GEF are a linear pair Prove: m/deg 1 m/gef 5 180º Make sure to carefully read the name of the angle whose measure you want to know. G D E F 18 Chapter Introduction to Proof

b. If /ABD and /DBC are complementary, then BA ' Given: /ABD and /DBC are complementary Prove: BA ' BC BC. A B D C c. If / and /3 are vertical angles, then / > /3. Given: / and /3 are vertical angles Prove: / > /3 3. Special Angles and Postulates 183

Problem 3 The terms postulate and theorem are defined. Euclid s first five postulates and Euclid s Elements are mentioned as the foundation of Euclidean Geometry. Non- Euclidean Geometry is also mentioned. Students answer questions related to parallel lines and the interior angle measures of triangles in spherical geometry. The Linear Pair Postulate, the Segment Addition Postulate, and the Angle Addition Postulate are stated. Students use each postulate to complete statements and answer questions. Guiding Questions for Discuss Phase Who was Euclid and what is he known for? What is the key difference between Euclidean and non- Euclidean geometry? PROBLEM 3 Postulates and Theorems A postulate is a statement that is accepted without proof. A theorem is a statement that can be proven. The Elements is a book written by the Greek mathematician Euclid. He used a small number of undefined terms and postulates to systematically prove many theorems. As a result, Euclid was able to develop a complete system we now know as Euclidean geometry. Euclid s first five postulates are: 1. A straight line segment can be drawn joining any two points.. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn that has the segment as its radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn that intersect a third line in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (This postulate is equivalent to what is known as the parallel postulate.) Euclid used only the first four postulates to prove the first 8 propositions or theorems of The Elements, but was forced to use the fifth postulate, the parallel postulate, to prove the 9th theorem. The Elements also includes five common notions : 1. Things that equal the same thing also equal one another.. If equals are added to equals, then the wholes are equal. 3. If equals are subtracted from equals, then the remainders are equal. 4. Things that coincide with one another equal one another. 5. The whole is greater than the part. Greek mathematician Euclid is sometimes referred to as the Father of Geometry. It is important to note that Euclidean geometry is not the only system of geometry. Examples of non-euclidian geometries include hyperbolic and elliptic geometry. The essential difference between Euclidean and non-euclidean geometry is the nature of parallel lines. 184 Chapter Introduction to Proof

Another way to describe the differences between these geometries is to consider two lines in a plane that are both perpendicular to a third line. In Euclidean geometry, the lines remain at a constant distance from each other and are known as parallels. In hyperbolic geometry, the lines curve away from each other. To better understand spherical geometry, have students relate to their knowledge of using a globe. On a globe, the equator is close to an example of a great circle. Show that any line of longitude intersects the equator at a 90-degree angle. In elliptic geometry, the lines curve toward each other and eventually intersect. Spherical geometry, like its name implies, is a geometry that substitutes a sphere for a plane, which makes it different from plane geometry in significant ways. In spherical geometry, some topics that you have learned about, such as parallel lines and the sum of the interior angles of a triangle are very different. q p Grouping Have students complete Questions 1 through 4 with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Questions 1 through 4 Would the shortest distance between points be an arc in spherical geometry? Are lines infinite in spherical geometry? In spherical geometry, lines are defined as great circles of a sphere, which divide the sphere into two congruent hemispheres. 1. Which of the figures in the diagram shown p, q, or r appear to be spherical lines? Explain your reasoning. Figures q and r are spherical lines, because they are great circles. Figure p is not a line.. Describe some similarities and differences between lines in Euclidean geometry and lines in spherical geometry, given this diagram. In Euclidean geometry, lines are straight and extend infinitely in two directions. In spherical geometry, lines are great circles of a sphere. These lines appear to extend infinitely in two directions also, but a line in spherical geometry overlaps itself. r. Special Angles and Postulates 185

3. Do you think that parallel lines can exist in spherical geometry? Explain your reasoning. Parallel lines do not exist, because lines are defined as great circles. Any two different great circles of a sphere must intersect. 4. At how many points do different spherical lines intersect? Explain your reasoning. Because spherical lines are great circles, two different spherical lines intersect at point. In spherical geometry, triangles have different properties as well. Consider the spherical triangle shown. It has 3 right angles, so the sum of its interior angle measures is 708. Grouping Have students complete Questions 5 through 7 with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Questions 5 through 7 What other geometric figures can you think of that would have different properties in spherical geometry? What properties do you think they would have? The sides of a spherical triangle are always arcs of great circles. 5. Describe what you think are some similarities and differences between triangles in Euclidean geometry and spherical triangles, given this diagram. Answers will vary. The sum of the interior angle measures of a triangle in Euclidean geometry is 1808. Spherical triangles do not necessarily have interior angle measures whose sum is 1808. The sides of a Euclidean triangle are all straight line segments, and the sides of a spherical triangle are curved line segments. 6. What do you think is the greatest sum of the interior angle measures of a spherical triangle? What is the least sum? Explain your reasoning. Answers will vary. The sum of the interior angle measures of a spherical triangle is greater than 1808 and less than 5408. 7. What do you think would be the properties of a right spherical triangle? Explain your reasoning. A right spherical triangle could be a triangle with just one 90-degree angle. Using this textbook as a guide, you will develop your own system of geometry, just like Euclid. You already used the three undefined terms point, line, and plane to define related terms such as line segment and angle. Investigate the properties of lines on a sphere and spherical triangles by drawing on a ball, such as a basketball or soccer ball. Your journey continues with the introduction of three fundamental postulates: The Linear Pair Postulate The Segment Addition Postulate The Angle Addition Postulate 186 Chapter Introduction to Proof

Grouping Have students complete Questions 8 through 10 with a partner. Then have students share their responses as a class. You will use these postulates to make various conjectures. If you are able to prove your conjectures, then the conjectures will become theorems. These theorems can then be used to make even more conjectures, which may also become theorems. Mathematicians use this process to create new mathematical ideas. The Linear Pair Postulate states: If two angles form a linear pair, then the angles are supplementary. Guiding Questions for Share Phase, Questions 8 through 10 What is the hypothesis in the Linear Pair Postulate? What is the conclusion in the Linear Pair Postulate? Does the Linear Pair Postulate tell you the sum of the measures of the two angles forming the linear pair is equal to 180? What is the definition of supplementary angles? Does the definition of supplementary angles tell you the sum of the measures of the two angles forming the linear pair is equal to 180? What is the hypothesis in the Segment Addition Postulate? What is the conclusion in the Segment Addition Postulate? Would the Segment Addition Postulate hold true if point B was not located between points A and C? What is the hypothesis in the Angle Addition Postulate? What is the conclusion in the Angle Addition Postulate? Would the Angle Addition Postulate hold true if point D did not lie in the interior of angle ABC? 8. Use the Linear Pair Postulate to complete each representation. a. Sketch and label a linear pair. D E F G b. Use your sketch and the Linear Pair Postulate to write the hypothesis. /DEG and /GEF form a linear pair. c. Use your sketch and the Linear Pair Postulate to write the conclusion. /DEG and /GEF are supplementary angles. d. Use your conclusion and the definition of supplementary angles to write a statement about the angles in your figure. m/deg 1 m/gef 5 180º. Special Angles and Postulates 187

The Segment Addition Postulate states: If point B is on AC and between points A and C, then AB 1 BC 5 AC. 9. Use the Segment Addition Postulate to complete each representation. a. Sketch and label collinear points D, E, and F with point E between points D and F. D E F b. Use your sketch and the Segment Addition Postulate to write the hypothesis. Point E is on DF and between points D and F. c. Use your sketch and the Segment Addition Postulate to write the conclusion. DE 1 EF 5 DF d. Write your conclusion using measure notation. m DE 1 m EF 5 m DF 188 Chapter Introduction to Proof

The Angle Addition Postulate states: If point D lies in the interior of /ABC, then m/abd 1 m/dbc 5 m/abc. 10. Use the Angle Addition Postulate to complete each representation. a. Sketch and label /DEF with EG drawn in the interior of /DEF. D G E F b. Use your sketch and the Angle Addition Postulate to write the hypothesis. Point D lies in the interior of /ABC. c. Use your sketch and the Angle Addition Postulate to write the conclusion. m/deg 1 m/gef 5 m/def Be prepared to share your solutions and methods.. Special Angles and Postulates 189

Check for Students Understanding Given: AB ' AC, m/abc 5 44º, and CD bisects /ACB 1. Determine the measure of each angle in the diagram. A 908 1138 D 678 678 1138 38 38 1348 C 1348 38 38 1368 B 448 448 1368. List all of the definitions, theorems, or reasons you used to determine the measures of the angles in the diagram. 1. The sum of the measures of the interior angles of a triangle is equal to 180. Vertical angles are congruent 3. Definition of linear pair 4. Definition of bisect 5. Definition of perpendicular 6. Definition of right angle 7. Definition of supplementary angles 190 Chapter Introduction to Proof