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January 5, 2011

Learn how to construct parallel and perpendicular lines and explore the various angle relationships formed by intersecting lines. Practice problems included.

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January 5, 2011

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  1. January 5, 2011 • Write your homework in your agenda: Angle worksheet • Let’s practice constructing parallel and perpendicular lines. • Then, let’s discuss angles and their relationship to one another. Take out a piece of paper to work out a few problems.

  2. Parallel Lines and Transversals The angle relationships that are formed

  3. What am I learning today? Constructions What will I do to show that I learned it? Determine the measurements of any given angle created by the intersection of parallel lines by a transversal

  4. Parallel Lines and Transversals What would you call two lines which do not intersect? Parallel The symbol || is used to indicate parallel lines. AB || CD

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

  6. Parallel Lines and Transversals Transversal - A transversal is a line which intersects two or more lines in a plane. The intersected lines do not have to be parallel. Lines j, k, and m are intersected by line t. Therefore, line t is a transversal of lines j, k, and m.

  7. Parallel Lines and Transversals Identifying Angles - Alternate interior angles ___________________are on the interior of the two lines and on opposite sides of the transversal. 1 3 5 7 2 4 6 8 Alternate interior angles are:

  8. Parallel Lines and Transversals Identifying Angles - Alternate exterior angles ___________________are on the exterior of the two lines and on opposite sides of the transversal. 1 3 5 7 2 4 6 8 Alternate exterior angles are:

  9. Parallel Lines and Transversals Identifying Angles - Corresponding angles _____________________are on the corresponding side of the two lines and on the same side of the transversal. 1 3 5 7 2 4 Corresponding angles are: 6 8

  10. Parallel Lines and Transversals Identifying Angles - Vertical angles _______________are pairs of opposite congruent angles formed by intersecting lines. 1 3 5 7 2 4 Vertical angles are: 6 8

  11. Parallel Lines and Transversals Identifying Angles - Adjacent angles _________________are in the same plane and share a common vertex and a common side. 1 3 5 7 Adjacent angles are: 2 4 6 8

  12. Parallel Lines and Transversals Identifying Angles – Check for Understanding 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. 4 3 2 1 5 6 8 7 2. 2 and 10 are alternate interior angles. 9 10 False - The angles are corresponding angles on transversal p. 11 12 15 16 14 13

  13. Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 3. 3 and 5 are alternate interior angles. False – The angles are vertical angles created by the intersection of q and r. 4 3 2 1 5 6 8 7 4. 1 and 15 are alternate exterior angles. 9 10 11 12 15 16 14 13 True - The angles are alternate exterior angles on transversal p.

  14. Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 5. 6 and 12 are alternate interior angles. True – The angles are alternate interior angles on transversal q. 4 3 2 1 5 6 8 7 6. 11 and 12 are complementary interior angles. 9 10 11 12 15 16 14 13 False – The angles are supplementary.

  15. Parallel Lines and Transversals Identifying Angles – Check for Understanding Determine if the statement is true or false. If false, correct the statement. 7. 3 and 4 are alternate exterior angles. False – The angles are supplementary. 4 3 2 1 5 6 8 7 8. 16 and 14 are corresponding angles. 9 10 11 12 15 16 14 13 True – The angles are corresponding on transversal s.

  16. The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning. EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m5=120°. Using the Vertical Angles Congruence Theorem, m4=120°. Because 4 and 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m8 = 120°.

  17. ALGEBRA Find the value of x. m4 + (x+5)° 180° = 115° + (x+5)° 180° = Substitute 115° for m4. x + 120 = 180 x = 60 EXAMPLE 2 Use properties of parallel lines SOLUTION By the Vertical Angles Congruence Theorem, m4=115°. Lines aand b are parallel, so you can use the theorems about parallel lines. Consecutive Interior Angles Theorem Combine like terms. Subtract 120 from each side.

  18. Use the diagram at the right. 1. If m 1 = 105°, find m 4, m 5, and m 8. m 4 = m 5 = m 8 = 105° 105° 105° for Examples 1 and 2 GUIDED PRACTICE SOLUTION

  19. Use the diagram at the right. for Examples 1 and 2 GUIDED PRACTICE 2. If m 3 = 68° and m 8 = (2x + 4)°, what is the value of x? Show your steps.

  20. 180° m 7 + m 8 = m 7 m 3 = 68° Substitute 68° for m 7 and (2x + 4)for m 8. 68° + 2x + 4 = 180° 7 and 8 are supplementary. 72 + 2x = 180° m 3 = 2x = 108 x = 54 for Examples 1 and 2 GUIDED PRACTICE SOLUTION Corresponding Angles Combine like terms. Subtract 72 from each side. Divide each side by 2.

  21. January 6, 2011 • Write your homework in your agenda: Part Three: Problem Sets worksheet • Take out your homework and leave it on your desk. • Here’s your Warm-Up….

  22. WARM-UP… Parallel Lines are cut by a transversal to create parking spaces. Two angle measures are given. Determine the 8 angle measures and label the diagram. 3x + 2 + 2x – 4 = 180 x = 36.4 111.2 68.8 68.8 3x + 2 = 111.2 degrees 111.2 68.8 2x - 4 111.2 68.8 degrees =

  23. What am I learning today? Triangle Proportionality Theorem What will I do to show that I learned it? Use proportionality theorems to determine segment length

  24. If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally. If TU ║ QS, then

  25. Finding the length of a segment • AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?

  26. Steps: DC EC BD AE 4 EC 8 12 4(12) 8 6 = EC = = EC = So, the length of EC is 6.

  27. QUESTION What is the Triangle Proportionality Theorem?

  28. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. If , then TU ║ QS.

  29. Determining Parallels • Given the diagram, determine whether MN ║ GH. LM 56 8 = = MG 21 3 LN 48 3 = = NH 16 1 8 3 ≠ 3 1 MN is not parallel to GH.

  30. QUESTION What is the Converse of the Triangle Proportionality Theorem?

  31. If three parallel lines intersect two transversals, then they divide the transversals proportionally. • If r ║ s and s║ t and l and m intersect, r, s, and t, then

  32. In the diagram 1  2  3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU? Using Proportionality Theorems

  33. 9 ● TU = 15 ● 11

  34. QUESTION What is the Proportionality Theorem for Parallel Lines?

  35. In the diagram, QS || UT , RS = 4, ST = 6, and QU = 9. What is the length of RQ ? RQ RS = QU ST RQ 4 = 9 6 EXAMPLE 1 Find the length of a segment SOLUTION Triangle Proportionality Theorem Substitute. RQ = 6 Multiply each side by 9 and simplify.

  36. 1. Find the length of YZ . XW XY = YZ WV 44 36 = 35 YZ 315 YZ = 11 315 So length of YZ = ANSWER 11 GUIDED PRACTICE SOLUTION Triangle Proportionality Theorem Substitute. Simplify

  37. 2.Determine whetherPS || QR . 5 PQ RS 50 5 40 = = = = SN 9 PN 90 9 72 PQ RS = SN PN ANSWER PS || QR , So Because = PS is parallel to QR GUIDED PRACTICE SOLUTION

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