Unit 3: Perpendicular and Parallel Lines

Transcription

1 Unit : Perpendicular and Parallel Lines Day 1 Parallel Lines and Planes Objectives: SWBAT identify relationships between lines PARALLEL LINES- Lines that are coplanar and do not intersect. Lines that have the same slope PARALLEL PLANES- Planes that never intersect PERPENDICULAR LINES- Lines that intersect at 90 degree angles. SKEW LINES- Lines that are not on the same plane, and do not intersect.

2 Write the relationship between lines AB and CD. Perpenedicular (intersect Skew (on different Parallel Lines ( see at 90 degree angles). planes) parallel lines arrows) Examples 1. Find all planes parallel to plane SKLM S K Plane NQRT 2. Find all segments that intersect with MT M N L Q MS, ML, TR, NT T R. Find all segments parallel to MT SN S K. Find all segments skew to MT SK, NQ, QR, KQ, KL M N L Q 5. Find all planes parallel to MTRL Plane NQKS T R 6. Find all segments parallel to NT SM, KL, QR S K 7. Find all segments skew to KL NQ, TR, SN, MT M N L Q T R

3 Day 2 Parallel Lines and Transversals Objectives: SWBAT identify relationships between lines and identify angles formed by transversals. Corresponding Angles TRANSVERSAL- a line interesting two other lines CORRESPONDING ANGLES~ SAME SPOT When two parallel lines are cut by a transversal, then the angles in the same position a re corresponding angles. 1& 5 & 8 2 & 6 & 7 ALTERNATE EXTERIOR ANGLES~ OPPOSITE OUTSIDE When two parallel lines are cut by a transversal, then the angles on the outside of the parallel lines, and opposite sides of the transversal are Alternate Exterior Angles. Alternate Exterior Angles 2 & 8 1& 7 ALTERNATE INTERIOR ANGLES~ OPPOSITE INSIDE When two parallel lines are cut by a transversal, then the angles on the inside of the parallel lines, and opposite sides of the transversal are Alternate Interior Angles. Alternate Interior Angles & 6 & 5 CONSECUTIVE INTERIOR ANGLES~ SAME SIDE SUPPLEMENTARY When two parallel lines are cut by a transversal, then the angles on the inside of the parallel lines, and the same side of the transversal are Consecutive Interior Angle. Consec utive Interior Angles & 5 & 6

4 VERTICAL ANGLES LINEAR PAIR Opposite angles are congruent Adjacent angles that add up 180 degrees Examples 1. Find all the Corresponding Angles. A E, C G, B F, D H 2. Find all the Alternate Interior Angles. C A D B G E H F D E, C F. Find all the Alternate Exterior Angles. B G, A H C A D B G E H F. Find all the Consecutive Interior Angles. mc me 180, md mf 180

5 State the relationship between angle A and B Consec utive Interior Angles Alternate Exterior Angles None Alternate Interior Angles Correseponding Angles Linear Pair 11. Use the diagram to the right to identify the following angle relationships. a) <BAD and <HEF Alternate Exterior Angles b) <CAH and <BEF Consecutive Interior Angles c) <GEH and <DAE Corresponding Angles

6 12. Given the following diagram find the relationship of <A and <B Consecutive Interior Angles 1. Given the following map, describe the corner street angles of John s house, Georgia s House, and Philip s house. John & Georgia Vertical Angles Georgia & Philip Corresponding Angles Philip & John Alternate Exterior Angles

7 Day Parallel Lines and Transversals Angle Relationships Objectives: SWBAT prove and use results about parallel lines and transversals. Corresponding Angles CORRESPONDING ANGLES~ SAME SPOT When two parallel lines are cut by a transversal, then the angles in the same position a are corresponding angles. Those angles are Congruent to each other ALTERNATE EXTERIOR ANGLES~ OPPOSITE OUTSIDE When two parallel lines are cut by a transversal, then the angles on the outside of the parallel lines, and opposite sides of the transversal are Alternate Exterior Angles. Those angles are Congruent to each other. Alternate Exterior Angles ALTERNATE INTERIOR ANGLES~ OPPOSITE INSIDE When two parallel lines are cut by a transversal, then the angles on the inside of the parallel lines, and opposite sides of the transversal are Alternate Interior Angles. Those angles are Congruent to each other. Alternate Interior Angles 6 5 CONSECUTIVE INTERIOR ANGLES~ SAME SIDE SUPPLEMENTARY When two parallel lines are cut by a transversal, then the angles on the inside of the parallel lines, and the same side of the transversal are Consecutive Interior Angle. Those angles Are supplementary to each other. Consec utive Interior Angles m m5 180 m m6 180

8 Perpendicular Transversal Theorem If a line is perpendicular to one line, then it is perpendicular, to every line to every line to that one. Supplementary Angles: Adjacent angles that add up 180 degrees Vertical Angles: Opposite angles are congruent EXAMPLES: 1. Given m 2 = 65, find each measure, and tell which postulate you used. a. m 1 = 65 (Corresponding to <2) b. m = 115 (< are supplementary to <1, so <1 is 115, since < and < are Alternate Exterior Angles so they are congruent) c. m 5 = 65 (Alternate Exterior angles with <1) d. m = 115 (< are supplementary to <1, so <1 is 115)

9 Assume the lines are parallel to find the value of x. 2.. These are Consecutive Interior angles and are supplementary. 5x 25 x x x 200 x 25 These are Alt Exterior Angles and are congruent. 7 x x x 217 x 1. Given that DG JK, find the measure of GEL These angles are Alternate Interior Angles And are congruent. 5. Find x x10 2x 0 2x 50 x 25 These angles are Corresponding Angles And are congruent. mgel x 10 mgel mgel 90 x 5 x 29 x

10 6. Find the value of y 7. Find the value of z These are Linear Pair and are supplementary. These are Vertical Angles are congruent. 17y1 6y y y2y z z Find m A and m B They are Consecutive Interior Angles, and are supplementary. y 5 y 10 y 5 y y y 175 y 25 mb y 5 mb 25 5 mb95 ma y 10 ma ma85

11 9. Seventh Avenue runs perpendicular to both 1 st and 2 nd streets. However, Maple Avenue makes a 10 degree angle with 2 nd street. What is the measure of angle 1? Since 7 th is Perpendicular to 2 nd and 1 st Street, And 1 st and 2 nd are parallel. Given that, Maple Avenue is a transversal to two parallel Lines. <1 is a linear pair with the Correspond Angle of the 10 angle. m m 1 50 Angles that are Congruent Angles that are Supplementary Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Linear Pair Vertical Angles

12 Day 5 Proving Lines are Parallel Objectives: SWBAT prove that two lines are parallel PROVING PARALLEL LINES USING CONVERSES CORRESPONDING ANGLES CONVERSE POSTULATE If the Corresponding Angles are congruent, then the lines are parallel. J K ALTERNATE EXTERIOR ANGLES THEOREM CONVERSE J If the Alternate Exterior Angles are congruent, then the lines are parallel. K CONSECUTIVE INTERIOR ANGLES THEOREM CONVERSE If the Consecutive Interior Angles are supplementary, then the lines are parallel. ALTERNATE INTERIOR ANGLES THEOREM CONVERSE If the Alternate interior Angles are congruent, then the lines are parallel. J K

13 In each example, determine if the lines are parallel or not. Explain why or why not. Corresponding Angles Are Congruent 1. So the lines are parallel by Corresponding Angles Theorem Converse 2... Alt Ext Angles Are Congruent So the lines are parallel by Alt. Ex t. Theorem Converse Consecutive Int Angles Are Supp. So the lines are parallel by Consec. Int. Theorem Converse Vetical Angles Not Enough Info Alt Int Angles Are Congruent So the lines are parallel by Alt. Int. Theorem Converse Consec So Int Angles are Supp sin ce they are Not Enough Info Linear Pair Not Enough Info

14 It is possible to prove that lines j and k are parallel given that A B. If possible, state the theorem or postulate that justifies the answer Alt Int Angles Are Congruent So the lines are parallel by Alt. Int. Theorem Converse Find the value of x that will make the two lines parallel. Not Enough Info Yes Many Re asons a. b These are supplementary (also a Linear Pair). This angle is then congruent to the Alternate Interior angle, so they are parallel These angles are Congruent Corresponding Angles so then they are parallel lines. c. d. xx0 x 0 These are Alternate Exterior Angles. 5x10 2x 50 x 60 x 20 These are Alternate Interior Angles.

15 Transative Parallel Lines Theorem If a line is parallel to another line, then it will be parallel to all other lines. Perpendicular Parallel Theorem If two lines are perpendicular to one line then they are parallel. EXAMPLES: State the postulate used to conclude a b. Given: 1 2 Given: a c and b c. Given: a c, c b a b c Alternate Exterior Perpendicular Parallel Transitive Parallel Angles Converse Theorem Lines Theorem Theorem

16 Partitioning a Segment Objectives: SWBAT find a point on a line segment between two given points that divides into a specific ratio: Partition: To break a segment into smaller pieces Based on a specific ratio. RIGHT x, y Ratio: A fraction, or parts of a whole LEFT x, y Break the following segments into the ratios. 1. to 2 2. to 6. to 5. to 1

17 1. Find the coordinates of P along the directed line segment AB so that the ratio of AP to PB is to 1. Ratio Ratio: :1 Mental Picture of Ratio x Distance units y Distance units Move Units to the Right 8 12 x Coordinate Move 9 Units Up 79 y Coordinate 2 Answer,2 2. Find the coordinates of P along the directed line segment AB so that the ratio of AP to PB is to 7. Given that A( 2, 10) and B(8, 10). Ratio: Ratio : 7 10 Mental Picture of Ratio x Distance units Move Units to the Right 2 x Coordinate 1 y Distance units Move 6 Units Up 10 6 y Coordinate Answer 1,

18 . Find the coordinates of P along the directed line segment AB so that the ratio of AP to PB is 2 to. Given that A(9, ) and B( 1, 2). Ratio: Ratio 2 : 2 5 Mental Picture of Ratio x Distance units Move Units to the Right 1 x Coordinate y Move Distance 2 2 units Units Up Answer 1, 5 y Coordinate 2.8 or 1 5

19 . An 80 mile trip is represented on a gridded map by a directed line segment from point M(, 2) to point N(9, 1). What point represents 20 miles into the trip? Ratio: Ratio 20 : Re duce 1 Mental Picture of Ratio Move x Distance 9 6 units Units to the Right 2 y Distance units Move Units Up 2 y Coordinate 5 x Coordinate.5 or 9 2 Answer 9,5 2