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Parallel lines and transversals
1. The document discusses angles formed when lines are intersected by a transversal line. 2. It defines types of angles formed, including corresponding angles, alternate interior/exterior angles, and consecutive interior/exterior angles. 3. Examples are provided to demonstrate identifying angle types and using angle properties involving parallel lines cut by a transversal.
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Parallel lines and transversals
- 2. A Line ThatIntersects 2 Or MoreA Line That Intersects 2 Or More Lines At Different Points IsLines At Different Points Is Called A TransversalCalled A Transversal transversaltransversal
- 3. When This Happens,WhenThis Happens, 8 Angles Are Formed8 Angles Are Formed 11 22 33 44 55 66 77 88
- 4. 11 22 33 44 55 66 77 88 This Forms 2NeighborhoodsThis Forms 2 Neighborhoods
- 5. 11 22 33 44 55 66 77 88 Remember:Remember: Vertical And LinearPair AnglesVertical And Linear Pair Angles VerticalVertical
- 6.
- 7. 11 33 5577 22 44 66 88 These Angles AreThese Angles Are Called ConsecutiveCalled Consecutive Or Same Side AnglesOr Same Side Angles
- 8. 33 44 55 66 11 22 77 88 Interior AnglesInterior Angles (Between2 lines)(Between 2 lines) Exterior AnglesExterior Angles ((outside the lines)outside the lines)
- 9. Alternate Angles AreOnAlternate Angles Are On Different Sides Of The TransversalDifferent Sides Of The Transversal AndAnd From Different NeighborhoodsFrom Different Neighborhoods 11 22 33 44 55 66 77 88 Alternate ExteriorAlternate Exterior Angles 1 And 8Angles 1 And 8 Angles 2 And 7Angles 2 And 7 Alternate InteriorAlternate Interior Angles 3 And 6Angles 3 And 6 Angles 4 And 5Angles 4 And 5
- 10. 11 33 55 77 22 44 66 88 ConsecutiveIntConsecutive Int Angles 3 and 5Angles 3 and 5 Angles 4 and 6Angles 4 and 6 Consecutive ExtConsecutive Ext Angles 1 and 7Angles 1 and 7 Angles 2 and 8Angles 2 and 8
- 11. 11 22 33 44 55 66 77 88 Corresponding Angles AreCorrespondingAngles Are Located In The Same Position InLocated In The Same Position In Each NeighborhoodEach Neighborhood
- 12. ANSWER alternate interior 1.5, 7 2. 3, 6 3. 1, 8 ANSWER alternate exterior ANSWER corresponding Identify the type of angles.
- 13. 1111 1212 1313 1414 15151616 1717 1818 Name The AnglesName The Angles 1.1. 11 and 1511 and 15 2.2. 12 and 1612 and 16 3.3. 13 and 1613 and 16 4.4. 12 and 1812 and 18 5.5. 14 and 1614 and 16 6.6. 14 and 1814 and 18 7.7. 11 and 1411 and 14 8.8. 15 and 1715 and 17
- 14. 1.1.CorrespondingCorresponding 2.2.CorrespondingCorresponding 3.3.Alt InteriorAlt Interior 4.4.Consecutive(SS) ExteriorConsecutive (SS) Exterior 5.5.Consecutive (SS) InteriorConsecutive (SS) Interior 6.6.CorrespondingCorresponding 7.7.VerticalVertical 8.8.Linear PairLinear Pair
- 15. 11 22 33 44 5566 77 88 99 1010 1111 1212 1313 1414 1515 1616 With This Diagram, We Can Work WithWith This Diagram, We Can Work With Angles In Different Neighborhoods As LongAngles In Different Neighborhoods As Long As They Are Connected By A TransversalAs They Are Connected By A Transversal Name the anglesName the angles 1.1. 1 and 31 and 3 2.2. 7 and 127 and 12 3.3. 11 and 1411 and 14 4.4. 6 and 106 and 10 5.5. 13 and 513 and 5 6.6. 9 and 69 and 6 7.7. 1 and 131 and 13 8.8. 5 and 45 and 4 9.9. 7 and 117 and 11 10.10. 6 and 116 and 11
- 16. 1.1. CorrespondingCorresponding 2.2. Alt.Int.Alt. Int. 3.3. Alt. Int.Alt. Int. 4.4. Cons. (SS) Int.Cons. (SS) Int. 5.5. CorrespondingCorresponding 6.6. Alt. Int.Alt. Int. 7.7. Consecutive ExtConsecutive Ext 8.8. Alt. ExtAlt. Ext 9.9. Cons. (SS) Int.Cons. (SS) Int. 10.10.NoneNone
- 17. 2. 2 and10 are alternate interior angles. Parallel Lines and Transversals ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 Determine if the statement is true or false. If false, correct the statement. 1. Line r is a transversal of lines p and q. ∠
- 18. 4. < 1and < 15 are alternate exterior angles. 3. <3 and < 5 are alternate interior angles. Parallel Lines and Transversals ∠ ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 Determine if the statement is true or false. If false, correct the statement. ∠ ∠
- 19. 6. < 10and < 11 are consecutive interior angles. 5. < 6 and < 12 are alternate interior angles. Parallel Lines and Transversals ∠ ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 Determine if the statement is true or false. If false, correct the statement. ∠ ∠
- 20. Determine if thestatement is true or false. If false, correct the statement. 7. < 3 and < 4 are alternate exterior angles. 8. < 16 and < 14 are corresponding angles. Parallel Lines and Transversals ∠ ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 ∠ ∠
- 21. 2. < 2and < 10 are alternate interior angles. Parallel Lines and Transversals Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 Determine if the statement is true or false. If false, correct the statement. 1. Line r is a transversal of lines p and q. True – Line r intersects both lines in a plane. False - The angles are corresponding angles on transversal p.
- 22. 4. < 1and < 15 are alternate exterior angles. 3. < 3 and < 5 are alternate interior angles. Parallel Lines and Transversals ∠ ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 Determine if the statement is true or false. If false, correct the statement. False – The angles are vertical angles created by the intersection of q and r. ∠ True - The angles are alternate exterior angles on transversal p. ∠
- 23. 6. < 10and < 11 are consecutive interior angles. 5. < 6 and < 12 are alternate interior angles. Parallel Lines and Transversals ∠ ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 Determine if the statement is true or false. If false, correct the statement. True – The angles are alternate interior angles on transversal q. ∠ True – The angles are consecutive interior angles on transversal s. ∠
- 24. Determine if thestatement is true or false. If false, correct the statement. 7. < 3 and < 4 are alternate exterior angles. 8. < 16 and < 14 are corresponding angles. Parallel Lines and Transversals ∠ ∠ Identifying Angles – Check for Understanding 1 2 3 4 5678 9 10 11 12 13141516 ∠ ∠ False– The angles are adjacent angles on transversal r. True – The angles are corresponding angles on transversal s.
- 26. If 2 ParallelLines Are Cut By AIf 2 Parallel Lines Are Cut By A Transversal Then:Transversal Then: Corresponding AnglesCorresponding Angles Are CongruentAre Congruent Alternate InteriorAlternate Interior Angles Are CongruentAngles Are Congruent Same Side Interior AnglesSame Side Interior Angles Are SupplementaryAre Supplementary
- 27. Remember ……… Even WithoutParallel Lines Vertical Angles Are Always Congruent Linear Pairs Are Always Supplementary
- 28. Example 1 SOLUTION By theCorresponding Angles Postulate, m 5 = 120°. Using the Vertical Angles Congruence Theorem, m 4 = 120°. Because 4 and 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m 8 = 120°. The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning.
- 29. Example 2 Use thediagram. 1. If m 1 = 105°, find m 4, m 5, and m 8. Tell which postulate or theorem you use in each case. Vertical Angles Congruence Theorem. Corresponding Angles Postulate.m 5 =105° Alternate Exterior Angles Theoremm 8 =105° m 4 =105° ANSWER
- 30. 11 22 33 44 5566 77 88 aa bb a ba b m 1 = 105m 1 = 105 Find:Find: 1.1. 3 =3 = 2.2. 6 =6 = 3.3. 7 =7 = 4.4. 4 =4 = 5.5. 5 =5 = 7575 7575 7575 105105 105105
- 31. 11 22 33 44 5566 77 88 aa bb 63°63° 117°117° 119°119° 119°119° 119°119° 119°119° 61°61° 63°63° 63°63°
- 32. Use the diagram. Ifm 3 = 68° and m 8 = (2x + 4)°, what is the value of x? Show your steps. 2. Guided Practice m 3 = m 7 68 + 2x + 4 = 180 2x + 72 = 180 2x = 108 x = 54 m 7 + m 8 =180 ANSWER
- 33. Example 3 ALGEBRA Find thevalue of x. SOLUTION By the Vertical Angles Congruence Theorem, m 4 = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines. Consecutive Interior Angles Theoremm 4 + (x+5)° = 180° Substitute 115° for m 4.115° + (x+5)°= 180° Combine like terms.x + 120 = 180 Subtract 120 from each side.x = 60
- 34. a ba b2x+62x+6 3x-103x-10 5x-205x-20 2x-102x-10 2x+6 = 3x-102x+6 = 3x-10 6 = x – 106 = x – 10 16 = x16 = x 5x-20+2x-10 = 1805x-20+2x-10 = 180 7x-30 = 1807x-30 = 180 7x = 2107x = 210 x = 30x = 30 4x+254x+25 6x-156x-15 4x+25 = 6x-154x+25 = 6x-15 25 = 2x-1525 = 2x-15 40 = 2x40 = 2x 20 = x20 = x