2.6 Special Angles on Parallel Lines Objectives: I CAN define angle pairs made by parallel lines. I CAN solve problems involving parallel lines.
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1 2.6 Special Angles on Parallel Lines Objecives: I CAN define angle pairs made by parallel lines. I CAN solve problems involving parallel lines. 1
2 coplanar: poins on he same plane ransversal: a line ha crosses oher coplanar lines Examples: Counerexample: 2
3 Angle Relaionships How many angles are here? 8 Corresponding angles. 2 & 6 4 & 8 3 & 7 1 & 5 Alernae inerior angles 4 & 5 3 & Alernae exerior angles 1 & 8 2 & 7
4 corresponding angles: angles on same side of lines Examples: Counerexample: 4 7 4
5 alernae-inerior angles: angles inside lines on opposie sides of ransversal Examples: Counerexample: 4 6 5
6 alernae-exerior angles: angles ouside lines on opposie sides of ransversal Examples: Counerexample: 2 8 6
7 Example #1 Idenify angle relaionships. a) Corresponding b) Alernae Inerior c) Alernae Exerior 7
8 Invesigaion 1. Draw wo parallel lines. (Use he lines on your paper or boh sides of your ruler.) 2. Label he lines a and b. 3. Draw a ransversal ha inersecs he parallel lines. 4. Label he angles wih numbers, like he diagram o he righ. 5. Place a shee of pay paper over angles 1, 2, 3, and 4 and race hem on he paper. 6. Slide he pay paper down o he oher four angles. 7. Wha do you noice abou corresponding angles, alernae inerior angles, and alernae exerior angles? 8
9 C-3a: CA Conjecure Corresponding Angles Conjecure If wo parallel lines are cu by a ransversal, hen corresponding angles are congruen
10 Example #2 m 5 = 120 (Corresponding Angles Conjecure) m 8 = 120 (Verical Angles Conjecure) The measure of hree of he numbered angles is 120º. Idenify he angles. Explain your reasoning. m 4 = 120 (Corresponding Angles Conjecure) 10
11 C-3b: AIA Conjecure Alernae Inerior Angles Conjecure If wo parallel lines are cu by a ransversal, hen alernae inerior angles are congruen
12 C-3c: AEA Conjecure Alernae Exerior Angles Conjecure If wo parallel lines are cu by a ransversal, hen alernae exerior angles are congruen
13 C-3: Parallel Lines Conjecure If wo parallel lines are cu by a ransversal, hen corresponding angles are congruen, alernae inerior angles are congruen, and alernae exerior angles are congruen
14 Invesigaion 1. Draw wo inersecing lines on your paper. 2. Copy hese lines ono a piece of pay paper. 3. Are he wo ses of angles congruen? 4. Slide he op copy so ha he ransversal says lined up. 5. Trace he lines and he angles from your paper ono he pay paper. 6. Wha kinds of angles were formed? 7. Use your ruler o measure he disance beween he wo lines in hree differen places. Are he wo lines parallel? 8. Repea Seps 1-7 again using anoher piece of pay paper, bu his ime roae your pay paper around 180 degrees so ha he ransversal lines up again. Trace he lines and angles and mark congruen angles. Are he lines parallel? 14
15 C-4: Converse of he Parallel Lines Conjecure If wo lines are cu by a ransversal o form pairs of congruen corresponding angles, congruen alernae inerior angles, or congruen alernae exerior angles, hen he lines are parallel
16 Angle Relaionships m e = 108 (verical angles) m f = 72 ( ) m g = 108 (coresponding o 108) m h = 72 (180 m < g)
17 C-3b: AIA Conjecure Alernae Inerior Angles Conjecure Given: ppq Prove: ppq Saemens 1. Given Reasons 2. Corresponding Angles Conjecure 3. Verical Angles Conjecure 4. Transiive Prop 17
18 Proof of C-3c: AEA Conjecure Alernae Exerior Angles Conjecure Given: ppq Prove: ppq Saemens 1. Given Reasons 2. Corresponding Angles Conjecure 3. Verical Angles Conjecure 4. Transiive Prop 18
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