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Chapter 3.5 Proving Lines Parallel

Chapter 3.5 Proving Lines Parallel. Check.3.1 Prove two lines are parallel, perpendicular, or oblique using coordinate geometry. Check.4.7 Identify perpendicular planes, parallel planes, a line parallel to a plane, skew lines, and a line perpendicular to a plane.

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Chapter 3.5 Proving Lines Parallel

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  1. Chapter 3.5 Proving Lines Parallel Check.3.1 Prove two lines are parallel, perpendicular, or oblique using coordinate geometry. Check.4.7 Identify perpendicular planes, parallel planes, a line parallel to a plane, skew lines, and a line perpendicular to a plane. CLE 3108.3.1 Use analytic geometry tools to explore geometric problems involving parallel and perpendicular lines, circles, and special points of polygons. Check.4.21 Use properties of and theorems about parallel lines, perpendicular lines, and angles to prove basic theorems in Euclidean geometry (e.g., two lines parallel to a third line are parallel to each other, the perpendicular bisectors of line segments are the set of all points equidistant from the endpoints, and two lines are parallel when the alternate interior angles they make with a transversal are congruent).

  2. Review • List the 4 relationships between the angles shown at left. • What is the equation of a line? • How do you calculate slope? • If two lines are parallel, what do you know to be true? • If two lines are perpendicular, what do you know to be true? 1 2 3 4 5 6 8 7

  3. containing the point (5, –2) in point-slope form? A. B. C. D.

  4. What equation represents a line containing points (1, 5) and (3, 11)? A.y = 3x + 2 B.y = 3x – 2 C.y – 6 = 3(x – 2) D.y – 6 = 3x + 2

  5. Notes Quiz Complete the following problems 1. Page 124, 26 • Page 181, 16 • Page 181, 28 • Page 190, 28 • Page 200; 14 • Page 200; 38 4 points for writing problem 4 points for correct answer 8 points for showing correct steps

  6. A. Given 1  5, is it possible to prove that any of the lines shown are parallel? A. Yes; ℓ║ n B. Yes; m ║ n C. Yes; ℓ║ m D. It is not possible to prove any of the lines parallel.

  7. Find mZYNso that || . Show your work. m WXP = m ZYN Alternate exterior angles 11x – 25 = 7x + 35 Substitution 4x – 25 = 35 Subtract 7x from each side. 4x = 60 Add 25 to each side. x = 15 Divide each side by 4.

  8. Since mWXP = mZYN,WXP ZYNand || . Now use the value of x to findmZYN. mZYN = 7x + 35 Original equation = 7(15) + 35 x= 15 = 140 Simplify. Answer:mZYN = 140 Check Verify the angle measure by using the value of x to find mWXP. mWXP = 11x – 25 = 11(15) – 25 = 140

  9. Identify Parallel Lines • Determine which lines if any are parallel Linear Pair at P 180 – 103 = 77 P a 103 77 Congruent Corresponding Angles a || b b Q 77 Linear Pair at R 180 – 100 = 80 c 80 100 Alternate Interior angles not Congruent, b not parallel to c R If Lines are parallel consecutive Interior angles are supplementary 77+100 does not equal 180, so not parallel

  10. Solve problems with Parallel Lines • Find x and mRSU so that m|| n • For m|| n then corresponding angles are congruent and mRSU mSTV mRSU mSTV 8x + 4 = 9x – 11 -8x + 11 -8x + 11 15 = x mRSU = 8x +4 = 8(15) + 4 = 120 + 4 = 124 U R (8x +4) V (9x -11) S m T l n

  11. Practice Assignment • Page 209, 8 - 26 Even

  12. Page 209, 8 - 26 Even 8. r || s; Converse of Corresponding Angles Postulate 10. r || s; Alternate Interior Angles Converse 12. u || v; Consecutive Interior AnglesConverse 14. No lines can be proven ||. 16. 39; Alt. Ext. s Conv. 18. 63; Alt. Int.  s Conv. 20. 27; Vert.  s Thm and Consec. Int.  s Conv. 22. Yes; when two pieces are put together, they form a 90° angle. Two lines that are perpendicular to the same line are parallel. 24. Since the corners are right angles, each pair of opposite sides is perpendicular to the same line. Therefore, each pair of opposite sides is parallel.

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