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DG4GSP_897_02.qxd 12/20/06 1:07 PM Page 27 Lesson 2.6 • Special Angles on Parallel Lines In this activity you’ll discover relationships among the angles formed when you intersect parallel lines with a third line called a transversal. Investigation 1: Which Angles Are Congruent? Sketch Step 1 and point C, not on AB . In a new sketch, construct AB Step 2 through point C. Construct a line parallel to AB Step 3 . Drag points C and A to make sure Construct AC you attached the three lines correctly. C A B Step 1 C A B Steps 2 and 3 Step 4 Construct points D, E, F, G, and H as shown. Step 5 Measure the eight angles in your figure. (Remember that to measure angles you need to select three points on the angle, making sure the middle point is always the vertex.) F C D A G Investigate 1. When two parallel lines are cut by a transversal, the pairs of angles formed have specific names and properties. Drag point A or B and determine which angles stay congruent. . Describe how many of the Also drag the transversal AC eight angles you measured appear to be always congruent. E B H Step 4 2. Angles FCE and CAB are a pair of corresponding angles. a. List all the pairs of corresponding angles in your construction. b. Write a conjecture describing what you observe about corresponding angles (CA Conjecture). 3. Angles ECA and CAG are a pair of alternate interior angles. a. List all the pairs of alternate interior angles in your construction. b. Write a conjecture describing what you observe about alternate interior angles (AIA Conjecture). (continued) Discovering Geometry with The Geometer’s Sketchpad ©2008 Key Curriculum Press CHAPTER 2 27 DG4GSP_897_02.qxd 12/20/06 1:07 PM Page 28 Lesson 2.6 • Special Angles on Parallel Lines (continued) 4. Angles FCE and HAG are a pair of alternate exterior angles. a. List all the pairs of alternate exterior angles in your construction. b. Write a conjecture describing what you observe about alternate exterior angles (AEA Conjecture). 5. Combine the three conjectures you made in Questions 2–4 into a single conjecture about parallel lines that are cut by a transversal (Parallel Lines Conjecture). 6. Suppose, in a similar sketch, all you knew was that the angle pairs described above had the properties you observed. Could you be sure that the original pair of lines were parallel? Try to answer this question first without using the computer. Investigation 2: Is the Converse True? Sketch Step 1 Step 2 Step 3 In a new sketch, construct two lines that are not quite parallel. Intersect both lines with a transversal. F Measure all eight angles formed by the three lines. Add points if you need them. Move the lines until the pairs of angles match the conjectures you made in the previous investigation. C D G E A B H Investigate 1. Lines with equal slopes are parallel. To check if your lines are parallel, measure their slopes. Write a new conjecture summarizing your conclusions (Converse of the Parallel Lines Conjecture). EXPLORE MORE 1. Angles ECA and BAC in Step 4 in Investigation 1 are sometimes called consecutive interior angles. In a new sketch, find all pairs of consecutive interior angles and make a conjecture describing their relationship. 2. Angles FCD and HAG in that same figure are sometimes called consecutive exterior angles. Find pairs of consecutive exterior angles in the figure and make a conjecture describing their relationship. 3. You can use the Converse of the Parallel Lines Conjecture to construct parallel lines. Construct a pair of intersecting lines AB as shown. Select, in order, points C, A, and B, and choose and AC Transform ⏐ Mark Angle. Double-click point C to mark it as a center for rotation. You figure out the rest. Explain why this works. C B A 28 CHAPTER 2 Discovering Geometry with The Geometer’s Sketchpad ©2008 Key Curriculum Press