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LESSON 19.1 Name Angles Formed by Intersecting Lines Class Date 19.1 Angles Formed by Intersecting Lines Essential Question: How can you find the measures of angles formed by intersecting lines? Resource Locker Common Core Math Standards The student is expected to: Explore 1 G-CO.9 Exploring Angle Pairs Formed by Intersecting Lines When two lines intersect, like the blades of a pair of scissors, a number of angle pairs are formed. You can find relationships between the measures of the angles in each pair. Prove theorems about lines and angles. Mathematical Practices MP.3 Logic Language Objective Explain to a partner the differences among complementary angles, supplementary angles, linear pairs, and vertical angles. A Possible answer: ENGAGE Possible answer: Identify any angles with known angle measures. Look for pairs of vertical angles and linear pairs of angles. Find angles that are complementary (if any) and supplementary. Use these relationships between pairs of angles together with the known angle measures to find the missing measures. 4 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©MNI/ Shutterstock Essential Question: How can you find the measures of angles formed by intersecting lines? Using a straightedge, draw a pair of intersecting lines like the open scissors. Label the angles formed as 1, 2, 3, and 4. B 3 2 Use a protractor to find each measure. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo, making sure students understand that a multiplying effect is created by the long handle and the short “jaws.” Then preview the Lesson Performance Task. 1 Module 19 Angle Measure of Angle m∠1 120° m∠2 60° m∠3 120° m∠4 60° m∠1 + m∠2 180° m∠2 + m∠3 180° m∠3 + m∠4 180° m∠1 + m∠4 180° be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Lesson 1 933 gh “File info” made throu Date Class ersecting rmed by Int Name s Fo 19.1 Angle Lines ion: How Quest Essential G-CO.9 Prove IN1_MNLESE389762_U8M19L1 933 can you theorems find the about lines measures of angles formed by . and angles Explori 1 a pair tedge, draw Using a straigh 3, and 4. 1, 2, formed as answer: Possible Credits: ©MNI/ 4 1 3 cting lines forme pairs are like the open Lines HARDCOVER PAGES 753760 d. scissors. Label Turn to these pages to find this lesson in the hardcover student edition. the angles 2 ctor to find Use a protra of interse Resource Locker rsecting r of angle s, a numbe a pair of scissor each pair. blades of angles in res of the ct, like the n the measu lines interse When two relationships betwee You can find Explore lines? intersecting ed by Inte Pairs Form ng Angle y • Image g Compan m∠1 m∠2 t Publishin m∠3 m∠4 Harcour n Mifflin © Houghto tock Shutters re. each measu Angle m∠1 + m∠2 m∠2 + m∠3 m∠3 + m∠4 of Angle Measure 120° 60° 120° 60° 180° 180° 180° 180° m∠1 + m∠4 Lesson 1 933 Module 19 9L1 933 62_U8M1 ESE3897 IN1_MNL 933 Lesson 19.1 4/19/14 12:15 PM 4/19/14 12:13 PM You have been measuring vertical angles and linear pairs of angles. When two lines intersect, the angles that are opposite each other are vertical angles. Recall that a linear pair is a pair of adjacent angles whose non-common sides are opposite rays. So, when two lines intersect, the angles that are on the same side of a line form a linear pair. EXPLORE 1 Reflect 1. Exploring Angle Pairs Formed by Intersecting Lines Name a pair of vertical angles and a linear pair of angles in your diagram in Step A. Possible answers: vertical angles: ∠1 and ∠3 or ∠2 and ∠4 ; linear pairs: ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, or ∠1 and ∠4 INTEGRATE TECHNOLOGY 2. Make a conjecture about the measures of a pair of vertical angles. Vertical angles have the same measure and so are congruent. 3. Use the Linear Pair Theorem to tell what you know about the measures of angles that form a linear pair. The angles in a linear pair are supplementary, so the measures of the angles add Students have the option of exploring the activity either in the book or online. up to 180°. Explore 2 QUESTIONING STRATEGIES Proving the Vertical Angles Theorem How would you describe a pair of supplementary angles in the drawing of intersecting lines? adjacent angles The conjecture from the Explore about vertical angles can be proven so it can be stated as a theorem. The Vertical Angles Theorem If two angles are vertical angles, then the angles are congruent. 4 1 3 Which pair of angles, if any, are congruent in a drawing of two intersecting lines? Opposite angles are congruent. 2 ∠1 ≅ ∠3 and ∠2 ≅ ∠4 Follow the steps to write a Plan for Proof and a flow proof to prove the Vertical Angles Theorem. Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3 4 Module 19 1 3 2 934 EXPLORE 2 © Houghton Mifflin Harcourt Publishing Company You have written proofs in two-column and paragraph proof formats. Another type of proof is called a flow proof. A flow proof uses boxes and arrows to show the structure of the proof. The steps in a flow proof move from left to right or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the box. You can use a flow proof to prove the Vertical Angles Theorem. Proving the Vertical Angles Theorem QUESTIONING STRATEGIES How do you identify vertical angles? They are opposite angles formed by intersecting lines. Lesson 1 PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U8M19L1 934 Math Background 4/19/14 12:13 PM Students will work with special angles. Congruent angles have the same measure. A pair of angles is supplementary if their sum is 180°. A pair of angles is complementary if their sum is 90°. Adjacent angles share one side without overlapping. Vertical angles are the non-adjacent angles formed by intersecting lines. They lie opposite each other at the point of intersection. Angles Formed by Intersecting Lines 934 A AVOID COMMON ERRORS Complete the final steps of a Plan for Proof: Because ∠1 and ∠2 are a linear pair and ∠2 and ∠3 are a linear pair, these pairs of angles are supplementary. This means that m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180°. By the Transitive Property, m∠1 + m∠2 = m∠2 + m∠3. Next: A common error that students make in writing proofs is using the result they are trying to prove as an intermediate step in the proof. Remind students to be alert for this type of circular reasoning. Subtract m∠2 from both sides to conclude that m∠1 = m∠3. So, ∠1 ≅ ∠3. B Use the Plan for Proof to complete the flow proof. Begin with what you know is true from the Given or the diagram. Use arrows to show the path of the reasoning. Fill in the missing statement or reason in each step. CURRICULUM INTEGRATION ∠1 and ∠2 are a linear pair. The common error mentioned above is discussed in the study of logic; it is known as the logical fallacy of begging the question. This is not the same as the general use of the phrase to “beg the question,” which often means “to raise another question.” Given (see diagram) ∠1 and ∠2 are supplementary . Linear Pair Theorem m∠1 + m∠2 = 180° Def. of supplementary angles ∠1 and ∠3 are vertical angles. Given ∠2 and ∠3 are a linear pair. Given (see diagram) ∠1 ≅ ∠3 © Houghton Mifflin Harcourt Publishing Company Def. of congruence ∠2 and ∠3 are supplementary. Linear Pair Theorem m∠2 + m∠3 = 180° Def. of supplementary angles m∠1 = m∠3 m∠1 + m∠2 = m∠2 + m∠3 Subtraction Property of Equality Transitive Property of Equality Reflect 4. Discussion Using the other pair of angles in the diagram, ∠2 and ∠4, would a proof that ∠2 ≅ ∠4 also show that the Vertical Angles Theorem is true? Explain why or why not. Yes, it does not matter which pair of vertical angles is used in the proof. Similar statements and reasons could be used for either pair of vertical angles. 5. Draw two intersecting lines to form vertical angles. Label your lines and tell which angles are congruent. Measure the angles to check that they are congruent. A D E C Possible answer: B By the Vertical Angles Theorem, ∠AEC ≅ ∠DEB and ∠AED ≅ ∠CEB. Checking by measuring, m∠AEC = m∠DEB = 45° and m∠AED = m∠CEB = 135°. Module 19 935 Lesson 1 COLLABORATIVE LEARNING IN1_MNLESE389762_U8M19L1 935 Small Group Activity Have students review the different angle pairs and lesson illustrations. Have each student draw one example of an angle pair, choosing from: a linear pair, supplementary angles, complementary angles, and vertical angles. Students then pass their drawings to others, who list all the information they know about that type of angle pair. Have them keep passing the drawings for other students to add more information until the drawing comes back to the student who drew it. Then have students identify and draw the angle that is a complement of an acute angle, a supplement of an obtuse angle, and the supplement of a right angle. acute, acute, right 935 Lesson 19.1 4/19/14 12:13 PM Explain 1 Using Vertical Angles EXPLAIN 1 You can use the Vertical Angles Theorem to find missing angle measures in situations involving intersecting lines. Example 1 Cross braces help keep the deck posts straight. Find the measure of each angle. Using Vertical Angles QUESTIONING STRATEGIES 146° 5 6 If two lines intersect to form four angles and one angle is an 80-degree angle, how do you know there is another 80-degree angle? The 80-degree angle forms a linear pair with two 100-degree angles, and they each form a linear pair with the remaining angle, so that angle must measure 80 degrees. 7 ∠6 Because vertical angles are congruent, m∠6 = 146°. ∠5 and ∠7 From Part A, m∠6 = 146°. Because ∠5 and ∠6 form a linear pair , they are CONNECT VOCABULARY supplementary and m∠5 = 180° - 146° = 34° . m∠ Differentiate the word vertical, meaning up and down, and the idea of vertical angles, which share a vertex but don’t need to be in an up-and-down position. They are vertical in respect to one another. also forms a linear pair with ∠6, or because it is a 7 = 34° because ∠ 7 vertical angle with ∠5. Your Turn 6. The measures of two vertical angles are 58° and (3x + 4)°. Find the value of x. 58 = 3x + 4 18 = x 7. The measures of two vertical angles are given by the expressions (x + 3)° and (2x - 7)°. Find the value of x. What is the measure of each angle? x + 3 = 2x - 7 x + 10 = 2x 10 = x The measure of each angle is (x + 3)° = (10 + 3)° = 13°. Module 19 936 © Houghton Mifflin Harcourt Publishing Company 54 = 3x Lesson 1 DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U8M19L1 936 Graphic Organizers 06/10/14 7:42 AM Have students create a graphic organizer with a central oval titled Pairs of Angles with leader lines to boxes for Adjacent angles, Linear pairs, Vertical angles, Supplementary angles, and Complementary angles. Ask students to draw and label an example of each pair of angles and to write a definition of each type of angle pair. Angles Formed by Intersecting Lines 936 Explain 2 EXPLAIN 2 Recall what you know about complementary and supplementary angles. Complementary angles are two angles whose measures have a sum of 90°. Supplementary angles are two angles whose measures have a sum of 180°. You have seen that two angles that form a linear pair are supplementary. Using Supplementary and Complementary Angles Example 2 B F ∠AFC and ∠CFD are a linear pair formed by an intersecting line and ray, ‹ › → − ‾ , so they are supplementary and the sum of their measures is 180°. By the AD and FC diagram, m∠CFD = 90°, so m∠AFC = 180° - 90° = 90° and ∠AFC is also a right angle. Because together they form the right angle ∠AFC, ∠AFB and ∠BFC are complementary and the sum of their measures is 90°. So, m∠AFB = 90° - m∠BFC = 90° - 50° = 40°. Find the measures of ∠DFE and ∠AFE. What fact do you use to find the angle measures in a linear pair? The angles in a linear pair are supplementary. solve for the measure of an angle in an angle-pair relationship.Students will write an equation to represent the angle-pair relationship and substitute the information that they are given to solve for the unknown. CONNECT VOCABULARY To help students remember the definitions of complementary and supplementary, explain that the letter c for complementary comes before s for supplementary, just as 90° comes before 180°. 937 Lesson 19.1 E Find the measures of ∠AFC and ∠AFB. QUESTIONING STRATEGIES ∠BFA and ∠DFE are formed by two so the angles are vertical intersecting lines and are opposite each other, angles. So, the angles are congruent. From Part A m∠AFB = 40°, so m∠DFE = 40° also. Because ∠BFA and ∠AFE form a linear pair, the angles are supplementary and the sum © Houghton Mifflin Harcourt Publishing Company INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Encourage students to write an equation to D 50° A supplementary angle might be. Encourage students to state that the angle measure is close to, but not exactly, 180°. Have them try to describe two lines that intersect to form these angles. How does an angle complementary to an angle A compare with an angle supplementary to A? It measures 90° less. Use the diagram below to find the missing angle measures. Explain your reasoning. C INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students what the maximum size of a What fact do you use to find the angle measures in a pair of complementary angles? Two angles are complementary if the sum of their measures is 90°. Using Supplementary and Complementary Angles of their measures is 180° . So, m∠AFE = 180° - m∠BFA = 180° - 40° = 140° . Reflect 8. In Part A, what do you notice about right angles ∠AFC and ∠CFD? Make a conjecture about right angles. Possible answer: Both angles have measure 90°. Conjecture: All right angles are congruent. Module 19 937 Lesson 1 LANGUAGE SUPPORT IN1_MNLESE389762_U8M19L1 937 4/19/14 12:13 PM Visual Cues Have students work in pairs to fill out an organizer like the following, starting with prior knowledge: Angle(s) Right Obtuse Acute Complementary Supplementary Definition Picture Your Turn ELABORATE You can represent the measures of an angle and its complement as x° and (90 - x)°. Similarly, you can represent the measures of an angle and its supplement as x° and (180 - x)°. Use these expressions to find the measures of the angles described. 9. The measure of an angle is equal to the measure of its complement. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 You may want to have students explore the 10. The measure of an angle is twice the measure of its supplement. x = 2(180 - x) x = 90 - x 2x = 90 x = 360 - 2x x = 45; so, 90 - x = 45 3x = 360 theorems in this lesson using geometry software. For example, have students construct a linear pair of angles, determine the measure of each angle, and use the software’s calculate tool to find the sum of the measures. Students can change the size of the angles and see that the measures change while their sum remains constant. x = 120; so, 180 - x = 60 The measure of the angle is 45° the measure of its complement is 45°. The measure of the angle is 120° the measure of its supplement is 60°. Elaborate 11. Describe how proving a theorem is different than solving a problem and describe how they are the same. Possible answer: A theorem is a general statement and can be used to justify steps in solving problems, which are specific cases of algebraic and geometric relationships. Both proofs and problem solving use a logical sequence of steps to justify a conclusion or to find a solution. QUESTIONING STRATEGIES 12. Discussion The proof of the Vertical Angles Theorem in the lesson includes a Plan for Proof. How are a Plan for Proof and the proof itself the same and how are they different? Possible answer: A plan for a proof is less formal than a proof. Both start from the given How many vertical angle pairs are formed where three lines intersect at a point? Explain. 6; if the small angles are numberered consecutively 1–6, the vertical angles are 1 and 4, 2 and 5, 3 and 6, and the adjacent angle pairs 1–2 and 4–5, 2–3 and 5–6, 3–4, and 6–1. information and reach the final conclusion, but a formal proof presents every logical step in detail, while a plan describes only the key logical steps. 13. Draw two intersecting lines. Label points on the lines and tell what angles you know are congruent and which are supplementary. Possible answer: K L J H Vertical angles are congruent: ∠GLK ≅ ∠JLH and ∠GLJ ≅ ∠KLH Angles in a linear pair are supplementary: ∠GLJ and ∠GLK; ∠GLJ and ∠JLH; ∠JLH and ∠HLK; and ∠HLK and ∠GLK © Houghton Mifflin Harcourt Publishing Company G SUMMARIZE THE LESSON Have students make a graphic organizer to show how some of the properties, postulates, and theorems in this lesson build upon one another. A sample is shown below. Angle Addition Postulate 14. Essential Question Check-In If you know that the measure of one angle in a linear pair is 75°, how can you find the measure of the other angle? The angles in a linear pair are supplementary, so subtract the known measure from 180°: Linear Pair Theorem 180 - 75 = 105, so the measure of the other angle is 105°. Module 19 938 Lesson 1 Vertical Angles Theorem IN1_MNLESE389762_U8M19L1 938 4/19/14 12:13 PM Angles Formed by Intersecting Lines 938 EVALUATE Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice Use this diagram and information for Exercises 1–4. Given: m∠AFB = m∠EFD = 50° Points B, F, D and points E, F, C are collinear. A Concepts and Skills Practice Explore Exploring Angle Pairs Formed by Intersecting Lines Exercise 1 Example 1 Proving the Vertical Angles Theorem Exercises 2–11 1. D Determine whether each pair of angles is a pair of vertical angles, a linear pair of angles, or neither. Select the correct answer for each lettered part. Vertical Linear Pair A. ∠BFC and ∠DFE B. ∠BFA and ∠DFE C. ∠BFC and ∠CFD D. ∠AFE and ∠AFC E. ∠BFE and ∠CFD F. ∠AFE and ∠BFC Exercises 12–14 2. Linear Pair Neither Vertical Linear Pair Neither Vertical Linear Pair Neither Vertical Linear Pair Neither Vertical Linear Pair Neither Find m∠AFE. 3. m∠EFB = m∠AFB + m∠AFE = 80° + 50° = 130° m∠AFE = 80° 4. Find m∠DFC. m∠DFC = m∠EFB, so m∠DFC = 130° 50° + m∠AFE + 50° = 180° © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Fresh Picked/Alamy Suggest still another way for students to remember which sum is 90 and which is 180: complementary and corner both start with the letter c, and corners are often right angles. Supplementary and straight both start with the letter s, and a straight line is a 180° angle. Neither Vertical m∠AFB + m∠AFE + m∠EFD = 180° CONNECT VOCABULARY C F E ASSIGNMENT GUIDE Example 2 Using Supplementary and Complementary Angles B Find m∠BFC. m∠BFC = m∠EFD = 50° 5. Represent Real-World Problems A sprinkler swings back and forth between A and B in such a way that ∠1 ≅ ∠2, ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. If m∠1 = 47.5°, find m∠2, m∠3, and m∠4. A 1 34 B 2 ∠1 ≅ ∠2, so m∠2 = 47.5° ∠1 and ∠3 are complementary, so m∠3 = 90 - 47.5 = 42.5° ∠2 and ∠4 are complementary, so m∠4 = 90 - 47.5 =42.5° Module 19 Exercise IN1_MNLESE389762_U8M19L1 939 939 Lesson 19.1 Lesson 1 939 Depth of Knowledge (D.O.K.) Mathematical Practices 1 1 Recall of Information MP.4 Modeling 2–4 1 Recall of Information MP.2 Reasoning 5–7 2 Skills/Concepts MP.2 Reasoning 8–11 1 Recall of Information MP.3 Logic 12–14 2 Skills/Concepts MP.3 Logic 15 2 Skills/Concepts MP.4 Modeling 9/23/14 7:50 AM Determine whether each statement is true or false. If false, explain why. 6. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 As students answer the questions using the If an angle is acute, then the measure of its complement must be greater than the measure of its supplement. False. The measure of an acute angle is less than 90°, so the measure of its complement will be less than 90° and the measure of its supplement will be greater than 90°. So, the measure of the supplement will be greater than the measure of the complement. 7. angle names in a pair of angles, encourage them to trace each angle along its letters and say the angle’s name aloud as they write it. This will help students get used to naming an angle with three letters. A pair of vertical angles may also form a linear pair. False. Vertical angles do not share a common side. 8. If two angles are supplementary and congruent, the measure of each angle is 90°. 9. If a ray divides an angle into two complementary angles, then the original angle is a right angle. True AVOID COMMON ERRORS True You can represent the measures of an angle and its complement as x° and (90 - x)°. Similarly, you can represent the measures of an angle and its supplement as x° and (180 - x)°. Use these expressions to find the measures of the angles described. Some students may have difficulty organizing the reasons or steps for a proof. You may wish to have students write the given reasons or steps on sticky notes. Then they can rearrange the sticky notes as needed to write the proof in the correct order. 10. The measure of an angle is three times the measure of its supplement. x = 3(180 - x) x = 540 - 3x 4x = 540 x = 135; so, 180 - x = 45 The measure of the angle is 135° and the measure of its supplement is 45°. 11. The measure of the supplement of an angle is three times the measure of its complement. 180 - x = 3(90 - x) 180 - x = 270 - 3x 2x = 90 © Houghton Mifflin Harcourt Publishing Company x = 45; so, 90 - x = 45 and 180 - x = 135 The measure of the angle is 45°, the measure of its complement is 45°, and the measure of its supplement is 135°. 12. The measure of an angle increased by 20° is equal to the measure of its complement. x + 20 = 90 - x 2x = 70 x = 35; so, 90 - x = 55 The measure of the angle is 35°, the measure of its complement is 55°. Module 19 Exercise IN1_MNLESE389762_U8M19L1 940 Lesson 1 940 Depth of Knowledge (D.O.K.) Mathematical Practices 16–17 3 Strategic Thinking MP.2 Reasoning 18–20 3 Strategic Thinking MP.3 Logic 4/19/14 12:12 PM Angles Formed by Intersecting Lines 940 Write a plan for a proof for each theorem. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Remind students that the shape of the letter X 13. If two angles are congruent, then their complements are congruent. Given: ∠ABC ≅ ∠DEF Prove: The complement of ∠ABC ≅ the complement of ∠DEF . Plan for Proof: If ∠ABC ≅ ∠DEF, then m∠ABC = m∠DEF. suggests the idea of vertical angles. The measure of the complement of ∠ABC = 90° - m∠ABC. The measure of the complement of ∠DEF = 90° - m∠DEF. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students of the definitions of vertical, Since m∠ABC = m∠DEF, the measure of the complement of ∠DEF = 90° - m∠ABC. Therefore, the measure of the complement of ∠ABC = the measure of the complements of ∠DEF. The measures of the complements of the angles are equal, so the complements of the angles are congruent. complementary, supplementary, and linear pairs. Have students use mental math to find complements and supplements of angles before using variables or expressions. 14. If two angles are congruent, then their supplements are congruent. Given: ∠ABC ≅ ∠DEF Prove: The supplement of ∠ABC ≅ the supplement of ∠DEF . Plan for Proof: If ∠ABC ≅ ∠DEF, then m∠ABC ≅ m∠DEF. The measure of the supplement of ∠ABC = 180° - m∠ABC. The measure of the supplement of ∠DEF = 180° - m∠DEF. Since m∠ABC = m∠DEF, the measure of the supplement of ∠DEF = 180° - m∠ABC. © Houghton Mifflin Harcourt Publishing Company Therefore, the measure of the supplement of ∠ABC = the measure of the supplement of ∠DEF. The measures of the supplements of the angles are equal, so the supplements of the angles are congruent. Module 19 IN1_MNLESE389762_U8M19L1 941 941 Lesson 19.1 941 Lesson 1 9/23/14 7:53 AM 15. Justify Reasoning Complete the two-column proof for the theorem “If two angles are congruent, then their supplements are congruent.” Statements Reasons 1. ∠ABC ≅ ∠DEF 1. Given 2. The measure of the supplement of ∠ABC = 180° - m∠ABC. 2. Definition of the 3. The measure of the supplement of ∠DEF = 180° - m∠DEF . 3. Definition of the supplement of an angle 4. m∠ABC = m∠DEF INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 If students have difficulty supplying the missing reasons in a proof, give them the reasons in random order and ask them to write the correct reason in the appropriate line of the proof. This type of scaffolding can be helpful for English language learners until they become more comfortable with the vocabulary of deductive reasoning. supplement of an angle 4. If two angles are congruent, their measures are equal. 5. The measure of the supplement of ∠DEF = 180° - m∠ABC. 5. Substitution Property of 6. The measure of the supplement of ∠ABC = the measure of the supplement of ∠DEF. 6. 7. The supplement of ∠ABC ≅ the supplement of ∠DEF . 7. If the measures of the supplements of two angles are equal, then supplements of the angles are congruent. Equality Substitution Property of Equality 16. Probability The probability P of choosing an object at random from a group of objects is found by the Number of favorable outcomes . Suppose the angle measures 30°, 60°, 120°, and 150° fraction P(event) = ___ Total number of outcomes are written on slips of paper. You choose two slips of paper at random. © Houghton Mifflin Harcourt Publishing Company a. What is the probability that the measures you choose are complementary? 1 possible pair (30°, 60°) 1 P(complementary) = ___ = _ 6 6 angle pairs total b. What is the probability that the measures you choose are supplementary? 2 possible pairs (30°, 150° and 60°, 120°) 2 =_ 1 P(supplementary) = ____ = _ 6 3 6 angle pairs total Module 19 IN1_MNLESE389762_U8M19L1 942 942 Lesson 1 4/19/14 12:12 PM Angles Formed by Intersecting Lines 942 PEER-TO-PEER DISCUSSION H.O.T. Focus on Higher Order Thinking Ask students to discuss with a partner the names and diagrams for various pairs of angles introduced in this lesson. Ask them to write and solve a word problem using complementary or supplementary angles on index cards, then switch cards with their partners to solve the problem. 17. Communicate Mathematical Ideas Write a proof of the Vertical Angles Theorem in paragraph proof form. 1 4 3 2 Given: ∠2 and ∠4 are vertical angles. Prove: ∠2 ≅ ∠4 In the diagram of intersecting lines, ∠2 and ∠4 are vertical angles. Also, ∠2 and ∠3 are a linear pair and ∠3 and ∠4 are a linear pair. By the Linear Pair Theorem, ∠2 and ∠3 are supplementary and ∠3 and ∠4 are supplementary. Then m∠2 + m∠3 = 180° and m∠3 + m∠4 = 180° by the definition of supplementary angles. By the Transitive Property of Equality, m∠2 + m∠3 = m∠3 + m∠4. Using the Subtraction Property of Equality, m∠2 = m∠4. So, ∠2 ≅ ∠4 by the definition of congruence. JOURNAL Have students write about each angle-pair relationship discussed in this lesson (supplementary angles, complementary angles, and vertical angles). Students should include definitions and drawings to support their descriptions. 18. Analyze Relationships If one angle of a linear pair is acute, then the other angle must be obtuse. Explain why. The sum of the measures of the two angles of a linear pair must be 180°. If the measure of one angle is less than 90°, then the measure of the other angle must be greater than 90°. 19. Critique Reasoning Your friend says that there is an angle whose measure is the same as the measure of the sum of its supplement and its complement. Is your friend correct? What is the measure of the angle? Explain your friend’s reasoning. Yes; 90°: the measure of its complement is 0°, and the measure of its supplement is 90°, so 0° + 90° = 90°. 20. Critical Thinking Two statements in a proof are: © Houghton Mifflin Harcourt Publishing Company m∠A = m∠B m∠B = m∠C What reason could you give for the statement m∠A = m∠C? Explain your reasoning. You could use either the Transitive Property of Equality, since both statements have m∠B in common; or you could use the Substitution Property of Equality by substituting m∠C for m∠B in the first statement. Module 19 IN1_MNLESE389762_U8M19L1 943 943 Lesson 19.1 943 Lesson 1 4/19/14 12:12 PM Lesson Performance Task CONNECT VOCABULARY The image shows the angles formed by a pair of scissors. When the scissors are closed, m∠1 = 0°. As the scissors are opened, the measures of all four angles change in relation to each other. Describe how the measures change as m∠1 increases from 0° to 180°. ∠3 ∠4 ∠2 A supplement is something that completes something else, as a vitamin supplement may complete a person’s daily vitamin requirement. The supplement of an angle completes a linear pair, with measures that complete a sum of 180°. A complement similarly brings something to completion, like a new coach who proves to be a perfect complement to the team. The complement of an angle brings a right angle to completion. ∠1 When m∠1 = 0°, m∠3 = 0°, and the measures of both ∠2 and ∠4 equal 180°. As m∠1 increases, the measure of ∠3 increases as well, so that m∠3 always equals m∠1. At the same time, the measures of ∠2 and ∠4 decrease steadily, both angles being congruent to each other and supplementary to ∠1. When m∠1 = 90°, the measures of the other three angles are also 90°. AVOID COMMON ERRORS When m∠1 > 90°, the measures of ∠2 and ∠4 are less than 90° and continually Students may have difficulty with the concept that as angle measures increase from 0° to 180°, their supplements change from obtuse to right to acute angles. Stress that supplementary angles have measures whose sum is 180°, and that one angle in a pair of supplementary angles that are not right angles will always be obtuse and one will be acute, and that supplements of right angles are right angles. decrease, reaching 0° when m∠1 = 180°. © Houghton Mifflin Harcourt Publishing Company Module 19 944 Lesson 1 EXTENSION ACTIVITY IN1_MNLESE389762_U8M19L1 944 A lever is a simple machine that enables the user to lift or move an object using less force than the weight of the object. A teeter-totter is an example of a “class 1” lever, which has a fulcrum between the object being moved and the applied force. Have students research “double class 1 levers,” of Class 1 Lever which pliers and scissors are examples. Students Force Load should explain how this type of lever differs from single class 1 levers, and describe how double class 1 levers create a mechanical advantage that allows Fulcrum users to produce forces greater than they apply. 4/19/14 12:12 PM Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Angles Formed by Intersecting Lines 944