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Crisscross Applesauce Angle Relationships Formed by Two Lines Intersected by a Transversal Learning Goals Key Terms In this lesson, you will: transversal alternate interior angles Explore the angles determined by two lines that are intersected by a transversal. Explore the measures of angles determined by two parallel lines that are intersected by a transversal. alternate exterior angles same-side interior angles same-side exterior angles Identify alternate interior angles. Identify alternate exterior angles. Identify same-side interior angles. Identify same-side exterior angles. Identify corresponding angles. Determine the measure of alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles. T ake two straws and lay them on your desk. Make them as close to parallel as © 2011 Carnegie Learning © 2011 Carnegie Learning you can. Then lay a third straw on top of the other two at any angle you like. Tape your entire construction together. Use your protractor to measure the angles you see. Notice anything interesting? Compare your constructions with your classmates’ constructions. What do you notice? 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 545 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 545 Problem 1 Transversal, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles are defined. Students sketch examples of these special pairs of angles and answer questions related to each pair of angles. Materials Protractor Straightedge Problem 1 Naming All the Angles In this lesson, you will explore all the angles that can be formed by transversals. A transversal is a line that intersects two or more lines. 1. Sketch an example of a transversal. 2. Compare your sketch with your classmates’ sketches. Did everyone sketch the same figure? Explain how the sketches are the same or different. No. Everyone did not sketch the same figure. Some students sketched more than three lines and some sketches did not contain any parallel lines. Each of the sketches shows at least three lines with at least one of the lines intersecting two or more lines. Grouping Have students complete Questions 1 through 8 with a partner. Then share the responses as a class. Alternate interior angles are angles formed when a line (transversal) intersects two other lines. These angles are on opposite sides of the transversal and are between the other two lines. 3. Sketch an example of alternate interior angles. 1 Share Phase, Questions 1 through 5 4. How many pairs of alternate interior angles are formed by two lines that are intersected by a transversal? Two pairs of alternate interior angles are formed by two lines that are intersected by a transversal. 5. Compare your sketch with your classmates’ sketches. Did everyone draw the same alternate interior angles? Explain how the sketches are the same or different. © 2011 Carnegie Learning Does a transversal always intersect two parallel lines? Explain. 2 angles formed have different measures, and some sketches do not contain parallel lines. Each of the sketches shows angles on opposite sides of the transversal and between the two other lines. 546 • Chapter 10 Line and Angle Relationships 546 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning No. Everyone did not draw the same alternate interior angles. The alternate interior Share Phase, Questions 6 through 8 • What is the difference between alternate interior and alternate exterior angles? Alternate exterior angles are angles formed when a line (transversal) intersects two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. 6. Sketch an example of alternate exterior angles. • What do alternate interior 1 and alternate exterior angles have in common? 2 7. How many pairs of alternate exterior angles are formed by two lines that are intersected by a transversal? Two pairs of alternate exterior angles are formed by two lines that are intersected by a transversal. 8. Compare your sketch with your classmates’ sketches. Did everyone draw the same alternate exterior angles? Explain how the sketches are the same or different. No. Everyone did not draw the same alternate exterior angles. The alternate exterior angles formed have different measures, and some sketches do not contain parallel lines. Each of the sketches shows angles on opposite sides of the transversal and outside the two other lines. Grouping other lines. These angles are on the same side of the transversal and are between the other two lines. 9. Sketch an example of same-side interior angles. © 2011 Carnegie Learning Have students complete Questions 9 through 17 with a partner. Then share the responses as a class. Same-side interior angles are angles formed when a line (transversal) intersects two 1 2 10. How many pairs of same-side interior angles are formed by two lines that are intersected by a transversal? © 2011 Carnegie Learning Two pairs of same-side interior angles are formed by two lines that are intersected by a transversal. 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 547 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 547 Share Phase, Questions 11 through 14 • What is the difference between same-side interior and same-side exterior angles? • What do same-side interior and same-side exterior angles have in common? 11. Compare your sketch with your classmates’ sketches. Did everyone draw the same angles? Explain how the sketches are the same or different. No. Everyone did not draw the same angles. The same-side interior angles formed have different measures, and some sketches do not contain parallel lines. Each of the sketches shows angles on the same side of the transversal and between the two other lines. Same-side exterior angles are angles formed when a line (transversal) intersects two other lines. These angles are on the same side of the transversal and are outside the other two lines. 12. Sketch an example of same-side exterior angles. 1 2 13. How many pairs of same-side exterior angles are formed by two lines that are intersected by a transversal? Two pairs of same-side exterior angles are formed by two lines that are intersected by a transversal. 14. Compare your sketch with your classmates’ sketches. Did everyone draw the same angles? Explain how the sketches are the same or different. No. Everyone did not draw the same angles. The same-side exterior angles formed have different measures, and some sketches do not contain parallel lines. Each of two other lines. Recall that corresponding angles are angles that have the same relative positions in geometric figures. 15. Sketch an example of corresponding angles. Include two lines intersected by a © 2011 Carnegie Learning the sketches shows angles on the same side of the transversal and outside the transversal in the sketch. 2 548 • Chapter 10 Line and Angle Relationships 548 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning 1 Share Phase, Questions 15 through 17 16. How many pairs of corresponding angles are formed by two lines that are intersected When a transversal intersects two lines, how many pairs of corresponding angles are formed? by a transversal? Four pairs of corresponding angles are formed by two lines that are intersected by a transversal. 17. Compare your sketch with your classmates’ sketches. Did everyone draw the same corresponding angles? Explain how the sketches are the same or different. No. Everyone did not draw the same corresponding angles. Some students drew corresponding angles with different measures, and some sketches did not contain parallel lines. Each of the sketches shows angles on the same side of the transversal, and one angle is located between the two lines and one angle is outside the two lines. Problem 2 Problem 2 Students are given three diagrams and will identify the transversals. Where Are the Transversals? 1. Suppose that <1 i <2, and both lines intersect <3. Identify the transversal(s). Grouping © 2011 Carnegie Learning Share Phase, Questions 1 through 3 • For a line to be considered 3 1 2 <3 is the transversal. © 2011 Carnegie Learning Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. a transversal, the line has to intersect how many other lines? 2. Suppose that ,1 i ,2, and both lines intersect ,3. Identify the transversal(s). 3 1 2 <1, <2, and <3 are all transversals. • If two coplanar non-parallel lines are not fully extended and you cannot see the point of intersection, do they still intersect? 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 549 • How many lines does ℓ2 intersect? • How many lines does ℓ3 intersect? • How many lines does ℓ1 intersect? 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 549 3. The arrowheads on these line segments indicate parallel relationships between opposite sides of the geometric figure. Transversals can be lines or line segments. Does this figure contain transversals? Explain your reasoning. Thisgeometricfigurecontainsfourtransversals.Eachsideofthefigure intersectstwootherlinesegments,soeachlinesegmentisatransversal. Street Map of Atlantic City, New Jersey Refer to the map of part of Atlantic City, New Jersey, to answer each question. Assume all line segments that appear to be perpendicular are perpendicular. Assume all line segments that appear to be parallel are parallel. MA NEW . AVE . AVE DEL RA MO MAR INE IRE MA PSH HAM NEW NT MO VER AND ISL DE RHO UT CTIC SEY NNE JER CO ARE AW AND RYL To New York & Philadelphia . AVE . AVE N R ITE P.O. . AVE . AVE P.O. IC IF PAC . AVE . AVE . AVE NTIC A ATL . AVE . AVE . AVE TIC ARC Light House . AVE . UNT MO FAIR . AVE 12 34 . AVE AVE . AVE . AVE . AVE . AVE . AVE Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. 56 78 . AVE MED . AVE . AVE Grouping EA RAN . AVE IA GIN ND VIR A STR A THE LIN ARO A N. C LIN ARO S. C SEE NES TEN K YOR NEW CKY TU KEN OIS ILLIN A IAN IND IO OH AN HIG MIC K AL DW AR BO N W E S Share Phase, Questions 1 and 2 • Does Atlantic Avenue intersect more than two other avenues? • How many distinct angles are formed when two lines intersect? • Is there more than one avenue you could have chosen? Explain. Atlantic City N.J. 1. Is Atlantic Ave. a transversal? Explain your reasoning. AtlanticAve.isatransversalbecauseitintersectsmorethantwootheravenues. 2. Locate the circle drawn on Atlantic Ave. This circle is drawn at the intersection of Atlantic Ave. and what other avenue? ThecircleisdrawnattheintersectionofAtlanticAve.andN.CarolinaAve. 550 • Chapter 10 Line and Angle Relationships 550 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning Students are given a street map of Atlantic City, N.J., and use the map to identify examples of transversals, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles. Problem 3 © 2011 Carnegie Learning Problem 3 Share Phase, Questions 3 through 5 • How many pairs of alternate 3. How many angles are formed at this intersection? Four angles are formed at this intersection. interior angles are formed? • How many pairs of alternate 4. Label each angle. exterior angles are formed? a. Place a 1 on the angle that would be considered the northwest angle. • How many pairs of same- b. Place a 2 on the angle that would be considered the northeast angle. c. Place a 3 on the angle that would be considered the southwest angle. side interior angles are formed? d. Place a 4 on the angle that would be considered the southeast angle. • How many pairs of same- 5. Using Atlantic Ave. and N. Carolina Ave., choose a third avenue such that Atlantic Ave. is a transversal. side exterior angles are formed? a. Label the four angles at this intersection /5, /6, /7, and /8 and describe the location of each angle (northeast, northwest, southeast, or southwest). • How many pairs of corresponding angles are formed? Answers will vary. Atlantic Ave. is a transversal between N. Carolina Ave. and New Jersey Ave. /5 is northwest, /6 is northeast, /7 is southwest, and /8 is southeast. b. List all pairs of alternate interior angles. /5, /4 and /2, /7 are pairs of alternate interior angles. c. List all pairs of alternate exterior angles. /1, /8 and /3, /6 are pairs of alternate exterior angles. d. List all pairs of same-side interior angles. © 2011 Carnegie Learning /2, /5 and /4, /7 are pairs of same-side interior angles. e. List all pairs of same-side exterior angles. /1, /6 and /3, /8 are pairs of same-side exterior angles. © 2011 Carnegie Learning f. List all pairs of corresponding angles. /1, /5; /2, /6; /3, /7; and /4, /8 are pairs of corresponding angles. 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 551 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 551 Problem 4 Students are given a street map of Washington, D.C., and use the map to identify examples of transversals, alternate interior angles, alternate exterior angles, same-side interior angles, same-side exterior angles, and corresponding angles. Problem 4 Washington, D.C., Map Use the map of Washington, D.C., to answer each question. Assume all line segments that appear to be parallel are parallel. Rhode Island Ave. New York Ave. Q St. 1 2 3 4 Grouping could be labeled ∠1? ∠2? ∠3? ∠4? • Did everyone place the label 6th St. 7th St. 9th St. New Jersey Ave. Massachusetts Ave. 1. Label /1, /2, /3, and /4 at the intersection of 7th St. and P St. 2. Label /5, /6, /7, and /8 at the intersection of 6th St. and P St. 3. Label /9, /10, /11, and /12 at the intersection of Massachusetts Ave. and P St. 4. Use a protractor to measure all 12 angles. 5. Consider only 6th St., 7th St., and P St. a. Which of these streets, if any, are transversals? P St. is a transversal. for ∠5 in the same position? • How many different positions could be labeled ∠5? ∠6? ∠7? ∠8? • Did everyone place the label b. Name the pairs of alternate interior angles. What do you notice about their angle measures? for ∠9 in the same position? /2 and /7 • How many different positions /4 and /5 could be labeled ∠9? ∠10? ∠11? ∠12? The alternate interior angles in each pair have equal measures. • How did you determine which street was a transversal? 552 • Chapter 10 Line and Angle Relationships • How did you locate the pairs of alternate interior angles? 552 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning • How many different positions P St. © 2011 Carnegie Learning for ∠1 in the same position? 9 10 11 12 N St. Have students complete Questions 1 through 6 with a partner. Then share the responses as a class. Share Phase, Questions 1 through 5, parts (a) and (b) • Did everyone place the label 5 6 7 8 Share Phase, Question 5, parts (c) through (g) • How did you locate the pairs of alternate exterior angles? c. Name the pairs of alternate exterior angles. What do you notice about their angle measures? /3 and /6 /1 and /8 • How did you locate the The alternate exterior angles in each pair have equal measures. corresponding angles? • How did you locate the same-side interior angles? d. Name the pairs of corresponding angles. What do you notice about their • How did you locate the angle measures? same-side exterior angles? • Does 6th Street intersect 7th /1 and /5 /2 and /6 Street? /3 and /7 /4 and /8 The corresponding angles in each pair have equal measures. e. Name the pairs of same-side interior angles. What do you notice about their angle measures? /2 and /5 /4 and /7 The same-side interior angles in each pair are supplementary. f. Name the pairs of same-side exterior angles. What do you notice about their angle measures? © 2011 Carnegie Learning /1 and /6 /3 and /8 The same-side exterior angles in each pair are supplementary. g. What is the relationship between 6th St. and 7th St.? © 2011 Carnegie Learning 6th St. is parallel to 7th St. 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 553 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 553 Share Phase, Question 6 How did you determine which of the three streets were transversals? 6. Consider only 6th Street, Massachusetts Avenue, and P Street. a. Which of these streets, if any, are transversals? 6th St., Massachusetts Ave., and P St. are all transversals. b. Name the pairs of alternate interior angles. What do you notice about their angle measures? /6 and /11 /8 and /9 The alternate interior angles in each pair do not have equal measures. c. Name the pairs of alternate exterior angles. What do you notice about their angle measures? /5 and /12 /7 and /10 The alternate exterior angles in each pair do not have equal measures. d. Name the pairs of corresponding angles. What do you notice about their angle measures? /5 and /9 /6 and /10 /7 and /11 /8 and /12 The corresponding angles in each pair do not have equal measures. e. Name the pairs of same-side interior angles. What do you notice about their angle measures? /6 and /9 /8 and /11 f. Name the pairs of same-side exterior angles. What do you notice about their angle measures? /5 and /10 /7 and /12 There is no special relationship. © 2011 Carnegie Learning There is no special relationship. g. What is the relationship between 6th St. and Massachusetts Ave.? 554 • Chapter 10 Line and Angle Relationships 554 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning 6th St. and Massachusetts Ave. are intersecting lines that are not parallel. Problem 5 Students use a protractor to measure special pairs of angles formed by a transversal intersecting two non-parallel lines and a transversal intersecting two parallel lines. They conclude that when two parallel lines are intersected by a transversal the alternate interior, alternate exterior, and corresponding angles are congruent. They also conclude that same-side interior and same-side exterior angles are supplementary. They also conclude that the transversal intersects two lines that are not parallel, these relationships do not hold true. Problem 5 Measuring Angles Formed by Two Lines and a Transversal 1. Draw a transversal intersecting two non-parallel lines, and number each angle. Then use a protractor to determine each angle measure. Answers will vary. 1 2 5 6 Use a straightedge. 3 4 7 8 3 1 2 m/1 5 76°, m/6 5 76°, m/5 5 104°, m/2 5 104°, m/3 5 115°, m/8 5 115°, m/4 5 65°, m/7 5 65°. 2. Draw a transversal intersecting two parallel lines, and number each angle.Then use a protractor to determine each angle measure. Answers will vary. Materials 1 5 2 6 3 Protractor 7 Straightedge 1 4 8 3 2 m/1 5 115°, m/6 5 115°, m/5 5 65°, m/2 5 65°, Have students complete Questions 1 through 9 with a partner. Then share the responses as a class. m/3 5 115°, m/8 5 115°, m/4 5 65°, m/7 5 65°. © 2011 Carnegie Learning Grouping Use the information from Questions 1 and 2 to answer Questions 3 through 8. 3. What do you notice about the measures of each pair of alternate interior angles when the lines are: a. non-parallel? © 2011 Carnegie Learning The alternate interior angles do not have equal measures. Share Phase, Questions 1 through 3 • What did you notice about the measures of each pair of vertical angles? b. parallel? The alternate interior angles have equal measures. 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 555 • Were any adjacent angles equal in measure? • Are there any perpendicular lines in your drawing? • Under what conditions are the alternate interior angles congruent? 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 555 Share Phase, Questions 4 through 7 • Under what conditions are the alternate exterior angles congruent? • Under what conditions are the corresponding angles congruent? • Under what conditions are the same-side interior angles supplementary? • Under what conditions are the same-side exterior angles supplementary? 4. What do you notice about the measures of each pair of alternate exterior angles when the lines are: a. non-parallel? The alternate exterior angles do not have equal measures. b. parallel? The alternate exterior angles have equal measures. 5. What do you notice about the measures of each pair of corresponding angles when the lines are: a. non-parallel? The corresponding angles do not have equal measures. b. parallel? The corresponding angles have equal measures. 6. What do you notice about the measures of the same-side interior angles when the lines are: a. non-parallel? The same-side interior angles are not supplementary. b. parallel? 7. What do you notice about the measures of the same-side exterior angles when the lines are: a. non-parallel? The same-side exterior angles are not supplementary. b. parallel? © 2011 Carnegie Learning The same-side interior angles are supplementary. 556 • Chapter 10 Line and Angle Relationships 556 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning The same-side exterior angles are supplementary. 8. Summarize your conclusions in the table by writing the relationships of the measures of the angles. The relationships are either congruent or not congruent, supplementary or not supplementary. Angles Two Parallel Lines Intersected by a Transversal Two Non-Parallel Lines Intersected by a Transversal Alternate Interior Angles congruent not congruent Alternate Exterior Angles congruent not congruent Corresponding Angles congruent not congruent Same-Side Interior Angles supplementary not supplementary Same-Side Exterior Angles supplementary not supplementary 9. Use your table in Question 8 to compare your conclusions with other groups or classmates. Also, compare the measures of the angles everyone used. What do you notice? Although we all had different measures for our angles, we had the same © 2011 Carnegie Learning © 2011 Carnegie Learning conclusions in our tables. 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 557 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 557 Problem 6 Students solve a “Who’s Correct?” problem which focuses on corresponding angles formed by two different transversals intersecting parallel lines. They will use their knowledge of special pairs of angles associated with a transversal intersecting two parallel lines to determine the unknown measures of angles in different situations. The third situation can be expressed as an equation and solved using addition and division. The last situation does require students to extend a transversal such that it intersects both parallel lines forming congruent alternate interior angles and a triangle. Problem 6 Solving for Unknown Angle Measures Sylvia and Scott were working together to solve the problem shown. ___ ___ Given: AB i CD . Solve for x. Show all your work. E xº A C 57º 57º B D 123 º 1. Sylvia concluded that x 5 66°. How did Sylvia get her answer? Sylvia assumed that the corresponding angles on ray ED were congruent to the angles formed on ray EC, so she solved for x by using the triangle at the top of the figure: 180° 2 57° 2 57° 5 66°. 2. Scott does not agree with Sylvia’s answer. He thinks there is not enough information to solve the problem. How could Scott alter the figure to explain his reason for disagreeing with Sylvia’s answer? Scott could redraw ray ED several different ways such that the measures of the Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. angles located at points B and D change. This would show Sylvia that the angles formed on ray EC are not congruent to the angles formed on ray ED. 3. Who is correct? Scott is correct. There is not enough information to solve for x. Share Phase, Questions 1 through 3 • How many transversals are in © 2011 Carnegie Learning Grouping • Can you assume the lengths of line segments EA and EB are equal? • If the length of line segment EA is not equal to the length of line segment EB, what does this tell you about the measure of ∠EAB and the measure of ∠EBA? 558 • Chapter 10 Line and Angle Relationships • Can you determine the measures of any angles formed at point B? Explain. • Can you determine the measures of any angles formed at point D? Explain. • Which angle measurements can you determine? 558 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning this diagram? Grouping Have students complete Questions 4 through 9 with a partner. Then share the responses as a class. 4. Opposite sides of this geometric figure are parallel. Suppose that the measure of angle M is equal to 30°. Solve for the measures of angles G, E, and O. Explain your reasoning. G Share Phase, Questions 4 through 6 • In Question 4, what is the relationship between ∠G and ∠M? E M O The measure of angle G is equal to 150° because angle M and angle G are same-side interior angles, so they are supplementary. The measure of angle E is 30° because angle G and angle E are same-side interior angles, so they are supplementary. The measure of angle O is 150° because angle E and angle O are same-side interior angles, so they are supplementary. • What is the measure of ∠G? • What is the relationship between ∠M and ∠O? • What is the measure of ∠O? • What is the relationship 5. Arrowheads indicate parallel lines. Determine the measures of all angles. between ∠O and ∠E? • What is the measure of ∠E? • What is the relationship 34° 146° 34° 146° 34° 146° 34° 146° between ∠E and ∠G? • What is the measure of ∠G? • In Question 5, which angles 6. Arrowheads indicate parallel lines. Determine the measures of all angles. in the diagram are equal to the measure of 34°? angle adjacent to the 34° angle? • In Question 6, how can the angles labeled x and x1100 be used to write an equation to solve for x? © 2011 Carnegie Learning 140º (x + 100)º 40º 140º 40º © 2011 Carnegie Learning • What is the measure of the 40º x º 140º 140º 40º x 1 (x 1 100) 5 180 2x 5 80, so x 5 40 • What is the sum of the measures of the two angles labeled x and x1100? 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 559 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 559 Share Phase, Questions 7 through 9 ‹___› ‹___› , ⊥ DE • In Question 7, if CE ___ ___› ___ ___› 7. In this figure, AB i CD and CE DE. Solve for x. Show all your work. what is the measure of ∠E? E 90° • How can you determine the 132° A measure of ∠ACD? 42° B 132° 48° C • How can you determine the 42° 42° D x° measure of ∠BDC? • In Question 8, will any triangles in the diagram help you to determine the measure of an angle? The value of x is 42. 8. Arrowheads indicate parallel lines, and boxes indicate that the angles are right angles. Determine the measure of each angle in this figure. 55° • What is the measure of the 44° angle adjacent to the 46° angle? • What does the little boxes in 44° 90° the diagram represent? 61° 29° 105° 75° 75° 105° • In Question 9, how can you extend a transversal so it forms a triangle in this diagram? • Is there a pair of alternate 134° 46° 46° 134° 134° 46° 46° 46° 134° 79° 105° 75° 75° 105° 29° 134° 46° 9. Solve for x. interior angles you could use to help solve this problem? 130° x 5 116 560 • Chapter 10 Line and Angle Relationships 560 • Chapter 10 Line and Angle Relationships © 2011 Carnegie Learning 64° 130° 50° 66° © 2011 Carnegie Learning xº Talk the Talk Students summarize the situations in which special pairs of angles are either congruent or supplementary. They will conclude that these situations exist when a transversal has intersected two parallel lines in most cases. Talk the Talk If two lines are intersected by a transversal… ● … when are alternate interior angles congruent? When two parallel lines are intersected by a transversal, alternate interior angles are congruent. ● … when are alternate exterior angles congruent? When two parallel lines are intersected by a transversal, alternate exterior angles Grouping Have students solve this problem with a partner. Then share the responses as a class. are congruent. ● … when are corresponding angles congruent? When two parallel lines are intersected by a transversal, corresponding angles are congruent. Share Phase, Talk the Talk • When are alternate interior ● … when are vertical angles congruent? When two parallel or non-parallel lines are intersected by a transversal, vertical angles not congruent? angles are congruent. • When are alternate exterior angles not congruent? ● • When are corresponding … when are same-side interior angles supplementary? When two parallel lines are intersected by a transversal, same-side interior angles are supplementary. angles not congruent? angles not supplementary? © 2011 Carnegie Learning • When are same-side interior ● • When are same-side exterior angles not supplementary? • When are adjacent angles not supplementary? • When are vertical angles not © 2011 Carnegie Learning congruent? …when are same-side exterior angles supplementary? When two parallel lines are intersected by a transversal, same-side exterior angles are supplementary. ● …when are adjacent angles supplementary? When two parallel or non-parallel lines are intersected by a transversal, adjacent angles are supplementary. Be prepared to share your solutions and methods. 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 561 10.3 Angle Relationships Formed by Two Lines Intersected by a Transversal • 561