Points, Lines, and Angles. m SAK = 130 K

Transcription

1 Points, Lines, and Angles Activity Exploring Angles m BAG = 42 m SAK = 130 K C D R B B P A G S A T A N Two rays with a common endpoint form an angle. Use Sketchpad to explore angles. 1. Open the sketch Angles 1 (Mac) or Angles1.gsp (Windows). 2. Notice BAG. It is an acute angle (an angle with measure less than 90 ). For each measure listed below, drag either B or G to observe the angle and make a sketch of your result. a. 20 b. 45 c d. 5 e f Notice SAK. It is an obtuse angle (an angle with measure between 90 and 180 ). For each measure listed below, drag S or K to observe the angle and make a sketch of your result. a. 120 b. 160 c. 95 d e. 135 f Of course, you can drag SAK to make it an acute angle as well. Drag S or K to observe an angle with each measure listed below and make a sketch of your result. a. 20 b A right angle measures exactly 90. Drag to make BAG and SAK right angles. Sketch your results below. a. m BAG = 90 b. m SAK = Unit 2 Points, Lines, and Angles Geometry Activities for Middle School Students with The Geometer s Sketchpad

2 Exploring Angles (continued) 6. Open the sketch Angles 2 (Mac) or Angles 2.gsp (Windows). 7. To measure an angle in Sketchpad, do the following steps: Hold down the Shift key. Click on three points: a point on either side, the vertex, and the point on the other side, in clockwise or counterclockwise order. Be sure the vertex is the middle point selected. Go to the Measure menu and choose Angle. You should see the measure of the angle appear on the screen. What kind of angle is RAN (acute, right, or obtuse)? What is the measure of RAN? You should have found m RAN = 70, which means it is an acute angle. 8. For each angle listed below, do the following: Write whether you think it is acute, right, or obtuse. Estimate the measure of the angle. Use Sketchpad to measure the angle, and record its actual measure. Change your answer for the kind of angle, if necessary. a. CAT b. PAT c. RAP d. CAP e. CAB f. DAN g. BAP h. TAD i. RAD j. BAR Acute, right, Estimated Actual Angle or obtuse Measure Measure Geometry Activities for Middle School Students with The Geometer s Sketchpad Unit 2 Points, Lines, and Angles 35

3 Points, Lines, and Angles Activity Exploring Special Pairs of Angles C M S H P A A T angle(cap) = 63 angle(hat) = 27 T angle(mit) = 114 I O angle(sok) = 66 K In geometry, you study several special kinds of pairs of angles. Use Sketchpad to explore some of these pairs. 1. Open the sketch Pairs 1 (Mac) or Pairs1.gsp (Windows). 2. Notice CAP and HAT. They are complementary angles. As you drag point X, the angles will remain complementary. What do you notice about the sum of their measures? 3. Use your observation from step 2 to define complementary angles. 4. Open the sketch Pairs 2 (Mac) or Pairs2.gsp (Windows). 5. Notice MIT and SOK. They are supplementary angles. As you drag point G, the angles will remain supplementary. What do you notice about the sum of their measures? 6. Use your observation from step 5 to define supplementary angles. 7. Open the sketch Pairs 3 (Mac) or Pairs3.gsp (Windows). 8. Notice that four angles are formed by a pair of intersecting lines. In this sketch, line BK intersects line GS to form the following angles: BAG, SAK, BAS, and GAK. 9. SAK and BAG are a pair of vertical angles. How do the measures of the angles compare? 36 Unit 2 Points, Lines, and Angles Geometry Activities for Middle School Students with The Geometer s Sketchpad

4 Exploring Special Pairs of Angles (continued) 10. BAS and GAK are also vertical angles. How do their measures compare? 11. Drag points S and K. As you drag, the pairs of angles remain vertical angles. What conclusion can you draw about the measures of any pair of vertical angles? 12. What kind of angles are BAS and SAK? What kind of angles are BAG and GAK? 13. From what you now know about vertical and supplementary angles, answer the following question: When two lines intersect, four angles are formed. How many angle measures would you need to know before you could name the measures of all the angles? Draw an example and write a short explanation to show how to do this. Geometry Activities for Middle School Students with The Geometer s Sketchpad Unit 2 Points, Lines, and Angles 37

5 Points, Lines, and Angles Activity Exploring Angles Formed by Parallel Lines and a Transversal When parallel lines are intersected by a third line, called a transversal, there are certain relationships among the angles formed. Use Sketchpad to explore these relationships. 1. Open the sketch Parallel (Mac) or Parallel.gsp (Windows). 2. Find HEX and EOY. They are a pair of corresponding angles. How do their measures compare? 3. For each angle given below, list its corresponding angle and tell how the measures compare. a. HEM b. MEO c. XEO 4. Drag point N slowly and notice the measures of each pair of corresponding angles. How do their measures always compare? Complete the following statement: If two parallel lines are cut by a transversal, then the corresponding angles are 5. Find MEO and EOY. They are a pair of alternate interior angles. How do their measures compare? 6. Name the other pair of alternate interior angles in the sketch: and. How do their measures compare? 38 Unit 2 Points, Lines, and Angles Geometry Activities for Middle School Students with The Geometer s Sketchpad

6 Exploring Angles Formed by Parallel Lines and a Transversal (continued) 7. Drag point N slowly and notice the measures of both pairs of alternate interior angles. How do their measures compare? Complete the following statement: If two parallel lines are cut by a transversal, then the alternate interior angles are 8. Find HEM and YON. They are a pair of alternate exterior angles. How do their measures compare? 9. Name the other pair of alternate exterior angles in the sketch: and. How do their measures compare? 10. Drag point N slowly and notice the measures of both pairs of alternate exterior angles. How do their measures always compare? Complete the following statement: If two parallel lines are cut by a transversal, then the alternate exterior angles are. Geometry Activities for Middle School Students with The Geometer s Sketchpad Unit 2 Points, Lines, and Angles 39