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Transcript

Properties of Parallel Lines Name(s): In this investigation, you’ll discover relationships among the angles formed when parallel lines are intersected by a third line, called a transversal. Sketch and Investigate 1. Construct dAB and point C, not on dAB. Select the line and the point; then, in the Construct menu, choose Parallel Line. 2. Construct a line parallel to dAB through point C. 3. Construct dCA. Drag points C and A to make sure the three lines are attached at those points. C C A Using the Text tool, click once on a point to show its label. Double-click the label to change it. To measure an angle, select three points, with the vertex your middle selection. Then, in the Measure menu, choose Angle. A B Step 1 B Steps 2 and 3 4. Construct points D, E, F, G, and H as shown at right. 5. Measure the eight angles in your figure. Be systematic about your measuring to be sure you don’t measure the same angle twice. F C D E 6. Drag point A or B and see G A B which angles stay congruent. Also drag the transversal dCA. H (Be careful not to change the point order on your lines. That Step 4 would change some angles into other angles.) Observe how many of the eight angles you measured appear to be always congruent. Q1 When two parallel lines are crossed by a transversal, the pairs of angles formed have specific names and properties. The chart on the next page shows one example of each type of angle pair. Fill in the chart with a second angle pair of each type, then state what relationship, if any, you observe between the angles in a pair type. Exploring Geometry with The Geometer’s Sketchpad © 2002 Key Curriculum Press Chapter 1: Lines and Angles • 17 Properties of Parallel Lines (continued) Angle Type Pair 1 Corresponding ∠FCE and ∠CAB Alternate interior ∠ECA and ∠CAG Alternate exterior ∠FCE and ∠HAG Same-side interior ∠ECA and ∠BAC Same-side exterior ∠FCD and ∠HAG Q2 Pair 2 One of the angle types has more than one pair. Name that angle type in the chart below, and name the third and fourth pairs of angles of that type. Angle Type Pair 3 Pair 4 7. Next, you’ll investigate the converses of your conjectures. In a new sketch, draw two lines that are not quite parallel. Construct a transversal. 8. Add points as needed, then measure all eight angles formed by the three lines. 9. Move the lines until you have two sets of four congruent angles. Q3 Relationship Relationship E F D C G A B H If two lines are crossed by a transversal so that corresponding angles, alternate interior angles, and alternate exterior angles are congruent, what can you say about the lines? Explore More 1. You can use the converse of the parallellines conjecture to construct parallel lines. Construct a pair of intersecting lines dAB and dAC as shown. Select, in order, points C, A, and B. Then, in the Transform menu, choose Mark Angle. Double-click point C to mark it as a center for rotation. You figure out the rest. Explain why this method works. 18 • Chapter 1: Lines and Angles C A B Exploring Geometry with The Geometer’s Sketchpad © 2002 Key Curriculum Press Properties of Parallel Lines (page 17) Prerequisites: Students should be familiar with angle notation and angle measurement. This activity introduces the terms transversal, corresponding angles, alternate interior angles, alternate exterior angles, same-side interior angles, and same-side exterior angles. Sketchpad Proficiency: Beginner Activity Time: 25–35 minutes Example Sketch: Parallel Lines.gsp Sketch and Investigate Q1 Students should complete the chart as shown below. Angle Type Pair 1 Pair 2 Rel’ship Corresponding ∠FCE , ∠CAB See Q2 below. Equal measure (or congruent) Alternate interior ∠ECA, ∠CAG ∠DCA, ∠CAB Equal measure Alternate exterior ∠FCE, ∠HAG ∠HAB, ∠DCF Equal measure Same-side interior ∠ECA, ∠BAC ∠DCA, ∠GAC Supplementary Same-side exterior ∠FCD, ∠HAG ∠FCE, ∠HAB Supplementary Q2 There are more than two pairs of corresponding angles. The three more pairs of corresponding angles not listed in the second row above are ∠ECA and ∠BAH, ∠HAG and ∠ACD, and ∠GAC and ∠DCF. Students should have recorded one of these pairs in the chart above. The relationship between angles in each pair is that they have equal measure. Q3 The lines must be parallel. Explore More 1. The last step is to rotate dAC by the marked angle. Because the alternate interior angles were constructed to be congruent, the new line will be parallel to dAB. Exploring Geometry with The Geometer’s Sketchpad © 2002 Key Curriculum Press Activity Notes—Chapter 1: Lines and Angles • 213