1 / 27

DOM 3.1

Explore special angle relationships, theorems, and angle properties in parallel lines and transversals. Learn about interior, exterior, corresponding, same-side, alternate interior and exterior angles.

gailh
Download Presentation

DOM 3.1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related