Angles and Lines in Geometry

Explore the concepts of parallel and skew lines, parallel planes, transversals, and angle relationships such as corresponding, alternate interior, and same-side interior angles. Learn to classify pairs of angles as either alternate interior, same-side interior, or corresponding. Practice identifying angles and lines in various geometric configurations.

• Geometry
• Angles
• Lines
• Parallel
• Skew

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Uploaded on Sep 30, 2024 | 1 Views

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Presentation Transcript

1. Parallel Lines Corresponding, Alternate Interior, & Same-side Interior Angles

2. Definitions Parallel Lines ( ) ___________________________________ ____________________________________________________ Skew Lines _______________________________________ ___________________________________________________ Parallel Planes ____________________________________ ___________________________________________________ Transversal ______________________________________ ____________________________________________________

3. Always, Sometimes, or Never? 1. Two lines in the same plane are _____________ parallel. 2. Two lines in the same plane are _____________ skew. 3. Two noncoplanar lines _____________ intersect. 4. Two planes _____________ intersect. 5. A line and a plane _____________ have exactly one point of intersection. 6. If two planes do not intersect, then they are _____________ parallel.

4. Name the two lines and the transversal that form each pair of angles k l 1. A.) 1 and 2 2 n 1 B.) 2 and 3 3 a 2. A.) 4 and 5 b 4 B.) 4 and 6 5 c d 6

5. Name the two lines and the transversal that form each pair of angles B C D 42 1. A.) 1 and 2 B.) 2 and 3 1 3 A E G 2. A.) 5 and 6 H 7 B.) 7 and 8 86 5 I F K J

6. x y Angles t 1 8 2 7 3 4 6 5 Exterior Angles Interior Angles Alternate Interior Angles Same-Side Interior Angles Corresponding Angles

7. Classify each pair of angles as alternate interior angles, same- side interior angles, or corresponding angles. 1. 2 and 4 2. 7 and 12 3. 10 and 11 4. 5 and 10 5. 14 and 15 a b 6. 3 and 11 c 3 7 4 8 2 1 5 6 9 10 11 15 12 d 13 14 16

8. Classify each pair of angles as alternate interior angles, same- side interior angles, or corresponding angles. 1. ??? and ??? U V W X 2. ??? and ??? S Y 3. ??? and ??? T Z Q P R 4. ??? and ??? 5. ??? and ??? 6. ??? and ??? 7. ??? and ??? 8. ??? and ???

9. 3-2 Properties of Parallel Lines

10. Properties of Parallel Lines 1. If two _________________ lines are cut by a _________________, then corresponding angles are _________________. 2. If two _________________ lines are cut by a _________________, then alternate interior angles are _________________. 3. If two _________________ lines are cut by a _________________, then same-side interior angles are _________________. 4. If a _________________ is perpendicular to one of two _________________ lines, then it is _________________ to the other one also.

11. State the postulate or theorem that justifies each statement. a b 1 3 4 1. 3 7 j 6 5 2 7 8 2. ? ? k 3. ? 6 + ? 7 = 180 4. 6 8 - (Arrowheads) Used to represent parallel lines

12. State the postulate or theorem that justifies each statement. a b 1 3 4 1. 4 6 j 6 5 2 7 8 2. 1 2 k 3. ? 5 = ? 7 4. 5 is supplementary to 8

13. Understanding Properties of Parallel Lines 1. Name seven angles that must be congruent to 1. 2. Name the eight angles that must be supplementary to 6. 3. If ? 5 = 60, what are the measures of the other numbered angles? 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16

14. Complete 1. If ? 2 = 80, then ? 6 = _______ and ? 7 = _______. 2. If ? 9 = 105, then ? 10 = _______ and ? 16 = _______. 3. If ? 8 = 85, then ? 16 = _______ and ? 10 = _______. 4. If ? 15 = 95, then ? 8 = _______ and ? 1 = _______. 12 11 9 4 1 2 10 3 16 15 13 8 5 14 7 6

15. Complete 1. If ? 3 = ? 4 + 30, find ? 5. 2. If ? 16 = ? 13 20, find ? 11. 12 11 9 4 1 2 10 3 16 15 13 45 5 14 7 6

16. Reminder! If two parallel lines are cut by a transversal, then Corresponding angles are congruent, Alternate Interior angles are congruent, and Same-Side Interior angles are supplementary If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.

17. Find the values of x and y. 1. 3y 6y 5x 2. 5y 8x 4x

18. Find the values of x and y. 3. 140 y (2x+10) 4. x 40 y 70

19. Find the values of x and y. 5. 65 x y 55 6. x 50 y 40

20. Find the values of x, y, and z. 7. y 30 x 42 6z 8. (3z+8) (4y+14) 70 x

21. Find the values of x, y, and z. 9. 5z 3x 60 (2y+10) 10. 2z 5y 40 x

22. 1. 1 8 k j 2. 4 6 3 4 5 3. 10 7 2 l 1 6 4. ? 3 + ? 4 = 180 8 7 n 9 10 5. 5 3 p 6. 6 7 In each exercise, some information is given. Name the lines (if any) that must be parallel. If there are no such lines, write none.

23. 1. 1 3 2. 1 4 P N 3. 2 5 1 2 3 4. 3 5 R O 4 5. 4 is supplementary to 5 5 T S In each exercise, some information is given. Name the lines (if any) that must be parallel. If there are no such lines, write none.

24. 1. 2. 3. j (3x+10) 6x (5x+15) 75 3x (5x-10) k j k j k Find the value of x that makes ? ?

25. B C A A B C 1. 2. (3x+2) (4x+5) (3y-1) (5y-7) (4x-18) (5x-13) F E D D E F Find the values of x and y that make ?? ?? and ?? ??