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DOM 3.1

DOM 3.1. Name the type of Angles. Ch 5.2 Parallel Lines & Transversals. Warm-Up: Give a Reason. Given: 1 & 2 are vertical s Conclusion: 1  2. If vertical s, then . (a) (b) (c) (d). Given: 1  2, 2  3 Conclusion: 1  3. If A  B and B  C, then A  C.

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DOM 3.1

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  1. DOM 3.1 Name the type of Angles

  2. Ch 5.2 Parallel Lines & Transversals

  3. Warm-Up: Give a Reason Given: 1 & 2 are vertical s Conclusion: 1  2 If vertical s, then  (a) (b) (c) (d) Given: 1  2, 2  3 Conclusion: 1  3 If A  B and B  C, then A  C Given: A and B are a linear pair Conclusion: A and B are supplementary If lin pair, then supp. Given: A and B are supplementary Conclusion: mA + mB = 180° If supp, then ms sum to 180°

  4. Parallel Lines and Transversals What would you call two lines which do not intersect? Parallel A solid arrow placed on two lines of a diagram indicate the lines are parallel. Exterior Interior The symbol || is used to indicate parallel lines. Exterior AB || CD

  5. Parallel Lines and Transversals A slash through the parallel symbol || indicates the lines are not parallel. AB || CD

  6. exterior interior exterior INTERIOR –The space INSIDE the 2 lines EXTERIOR -The space OUTSIDE the 2 lines

  7. 1 2 3 4 5 6 7 8 Special Angle Relationships Interior Angles <3 & <6 are Alternate Interior angles <4 & <5 are Alternate Interior angles <3 & <5 are Same Side Interior angles <4 & <6 are Same Side Interior angles Exterior Interior Exterior Exterior Angles <1 & <8 are Alternate Exterior angles <2 & <7 are Alternate Exterior angles <1 & <7 are Same Side Exterior angles <2 & <8 are Same Side Exterior angles

  8. 1 2 3 4 6 5 7 8 If the lines are not parallel, these angle relationships DO NOT EXIST. Special Angle RelationshipsWHEN THE LINES ARE PARALLEL ♥Theorem 3.4 Alternate Interior Angles are CONGRUENT ♥Theorem 3.6 Alternate Exterior Angles are CONGRUENT ♥Theorem 3.5 Same Side Interior Angles are SUPPLEMENTARY ♥Postulate 15 Corresponding Angles are CONGRUENT Exterior Interior Exterior

  9. Theorem 3.7 Perpendicular Transversal • If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. j h k j  k

  10. Corresponding Angles & Consecutive Angles Corresponding Angles: Two angles that occupy corresponding positions.  2  6, 1  5,3  7,4  8 Exterior 1 2 3 4 Interior 5 6 7 8 Exterior

  11. L Line L G Line M P A B Line N Q D E F Corresponding Angles When two parallel lines are cut by a transversal, pairs of corresponding angles are formed. ÐGPB = ÐPQE ÐGPA = ÐPQD ÐBPQ = ÐEQF ÐAPQ = ÐDQF Four pairs of corresponding angles are formed. Corresponding pairs of angles are congruent.

  12. Same Side Interior/Exterior Angles Same Side Interior Angles: Two angles that lie between parallel lines on the same sides of the transversal. Same Side Exterior Angles: Two angles that lie outside parallel lines on the same sides of the transversal. m3 +m5 = 180º, m4 +m6 = 180º Exterior 1 2 m1 +m7 = 180º, m2 +m8 = 180º 3 4 Interior 5 6 7 8 Exterior

  13. L Line L G Line M P A B Line N Q D E F Interior Angles The angles that lie in the area between the two parallel lines that are cut by a transversal, are calledinterior angles. Exterior ÐBPQ + ÐEQP = 1800 600 1200 Interior 1200 600 ÐAPQ + ÐDQP = 1800 Exterior The measures of interior angles in each pairadd up to 1800. A pair of interior angles lie on thesame sideof the transversal.

  14. Alternate Interior/Exterior Angles • Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair). Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal. 3  6,4  5 2  7,1  8 1 2 Exterior 3 4 Interior 5 6 7 8 Exterior

  15. L Line L G Line M P A B Line N Q D E F Alternate Interior Angles Alternate angles are formed on opposite sides of the transversal and at different intersecting points. ÐBPQ = ÐDQP ÐAPQ = ÐEQP Two pairs of alternate angles are formed. Pairs of alternate angles are congruent.

  16. 1 2 3 4 6 5 7 8 Let’s Practice m<1=120° Find all the remaining angle measures. 120° 60° 120° 60° 120° 60° 120° 60°

  17. Ex. 4: Using properties of parallel lines • Use the properties of parallel lines to find the value of x. 120° (x +15)°

  18. Another practice problem Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements. 40° 180-(40+60)= 80° 60° 80° 60° 40 80° 120° 100° 60° 80° 120° 60° 100°

  19. Example 2: Complete the Proof Given: p || q; m1 = 32° Prove: m2 = 32° Statement Reason p || q; m1 = 32° Given if parallel, then corresp. s  2  1 if , then = m2 = m1 if a = b, then subst a for b m2 = 32°

  20. Example 1: Proving the Alternate Interior Angles Theorem • Given: p ║ q • Prove: 1 ≅ 2 1 2 3

  21. Statements: p ║ q 1 ≅ 3 3 ≅ 2 1 ≅ 2 Reasons: Given Corresponding Angles Postulate Vertical Angles Theorem Transitive Property of Congruence Proof

  22. 2 Example 3: Solve for x p Given: p || q Prove: x = 63 q Statement Reason 1 p || q Given If parallel, then alt. ext. s  1  2 m1 = m2 If , then = If a = b, then subst a for b x – 8 = 55° If a = b, then a + c = b + c x = 63

  23. Example 2: Using properties of parallel lines • Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use. • A. m 6 B. m 7 • C. m 8 D. m 9 9 6 8 5 7

  24. Solutions: • m 6 = m 5 = 65° • Vertical Angles Theorem • m 7 = 180° - m 5 =115° • Linear Pair postulate • m 8 = m 5 = 65° • Corresponding Angles Postulate • m 9 = m 7 = 115° • Alternate Exterior Angles Theorem

  25. Closure • Given that m3 = 55, what is m1? Why?  2 Flag of Puerto Rico 3  1

  26. Line L G Line M P A B 500 Line L G 1300 Line M Line N A P B Q E D Line L G F Line N Q D E Line M P A B F Line N Q E D F Name the pairs of the following angles formed by a transversal. Test Yourself Corresponding angles Alternate angles Interior angles

  27. Assignment • #8-28 EVEN • #34-44 EVEN (pg. 146-148)

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