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Chapter 3

Chapter 3. Perpendicular and Parallel Lines. Chapter Objectives. Identify parallel lines Define angle relationships between parallel lines Develop a Flow Proof Use Alternate Interior, Alternate Exterior, Corresponding, & Consecutive Interior Angles

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Chapter 3

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  1. Chapter 3 Perpendicular and Parallel Lines

  2. Chapter Objectives • Identify parallel lines • Define angle relationships between parallel lines • Develop a Flow Proof • Use Alternate Interior, Alternate Exterior, Corresponding, & Consecutive Interior Angles • Calculate slopes of Parallel and Perpendicular lines

  3. Lesson 3.1 Lines and Angles

  4. Lesson 3.1 Objectives • Identify relationships between lines. • Identify angle pairs formed by a transversal. • Compare parallel and skew lines.

  5. Lines and Angle Pairs Alternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal. Transversal 2 1 3 4 Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal. 5 6 Corresponding Angles – because they lie in corresponding positions of each intersection. 8 7 Alternate Interior Angles – because they lie inside the two lines andon opposite sides of the transversal.

  6. Example 1 Determine the relationship between the given angles • 3 and 9 • Alternate Interior Angles • 13 and 5 • Corresponding Angles • 4 and 10 • Alternate Interior Angles • 5 and 15 • Alternate Exterior Angles • 7 and 14 • Consecutive Interior Angles

  7. Parallel versus Skew • Two lines are parallel if they are coplanar and do not intersect. • Lines that are not coplanar and do not intersect are called skew lines. • These are lines that look like they intersect but do not lie on the same piece of paper. • Skew lines go in different directions while parallel lines go in the same direction.

  8. Example 2 Complete the following statements using the words parallel, skew, perpendicular. • Line WZ and line XY are _________. • parallel • Line WZ and line QW are ________. • perpendicular • Line SY and line WX are _________. • skew • Plane WQR and plane SYT are _________. • parallel • Plane RQT and plane WQR are _________. • perpendicular • Line TS and line ZY are __________. • skew • Line WX and plane SYZ are __________. • parallel.

  9. Parallel and Perpendicular Postulates:Postulate 13-Parallel Postulate • If there is a line and a point not on the line, then there is exactly one line through the point that is parallel to the given line.

  10. Parallel and Perpendicular Postulates:Postulate 14-Perpendicular Postulate • If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

  11. Homework 3.1 • In Class • 2-9 • p132-135 • Homework • 10-31 • Due Tomorrow

  12. Lesson 3.2 Proof and Perpendicular Lines

  13. Lesson 3.2 Objectives • Develop a Flow Proof • Prove results about perpendicular lines • Use Algebra to find angle measure

  14. 5 7 6 Flow Proof • A flow proof uses arrows to show the flow of a logical argument. • Each reason is written below the statement it justifies.  6 and  7 are a linear pair • 5 and  6 are a linear pair GIVEN: 5 and  6 are a linear pair  6 and  7 are a linear pair PROVE:  5  7 Given Given  5 and  6 are supplementary  6 and  7 are supplementary Linear Pair Postulate Linear Pair Postulate  5  7 Congruent Supplements Theorem

  15. g h Theorem 3.1:Congruent Angles of a Linear Pair • If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. So g  h

  16. Theorem 3.2:Adjacent Angles Complementary • If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

  17. Theorem 3.3: Four Right Angles • If two lines are perpendicular, then they intersect to form four right angles.

  18. Homework 3.2 • None! • Move on to Lesson 3.3

  19. Lesson 3.3 Parallel Lines and Transversals

  20. Lesson 3.3 Objectives • Prove lines are parallel using tranversals. • Identify properties of parallel lines.

  21. 1 2 3 4 5 6 8 7 Postulate 15:Corresponding Angles Postulate • If two parallel lines are cut by a transversal, then corresponding angles are congruent. • You must know the lines are parallel in order to assume the angles are congruent.

  22. 1 2 3 4 5 6 8 7 Theorem 3.4:Alternate Interior Angles • If two parallel lines are cut by a transversal, then alternate interior angles are congruent. • Again, you must know that the lines are parallel. • If you know the two lines are parallel, then identify where the alternate interior angles are. • Once you identify them, they should look congruent and they are.

  23. 1 2 3 4 5 6 8 7 Theorem 3.5:Consecutive Interior Angles • If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. • Again be sure that the lines are parallel. • They don’t look to be congruent, so they MUST be supplementary. 180o = + + = 180o

  24. 1 2 4 3 8 7 5 6 Theorem 3.6:Alternate Exterior Angles • If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. • Again be sure that the lines are parallel.

  25. Theorem 3.7:Perpendicular Transversal • If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. • Again you must know the lines are parallel. • That also means that you now have 8 right angles!

  26. Example 3 Find the missing angles for the following: 120o 120o 60o 120o 140o 140o 105o 70o 110o 105o 110o

  27. Homework 3.3 • In Class • 3-6 • p146-149 • Homework • 8-26, 34-44 even • Due Tomorrow • Quiz Wednesday • Lessons 3.1-3.3 • Emphasis on 3.1 & 3.3

  28. Lesson 3.4 Proving Lines are Parallel

  29. Lesson 3.4 Objectives • Prove that lines are parallel • Recall the use of converse statements

  30. 1 2 3 4 5 6 8 7 Postulate 16:Corresponding Angles Converse • If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. • You must know the corresponding angles are congruent. • It does not have to be all of them, just one pair to make the lines parallel.

  31. 1 2 4 3 5 6 8 7 Theorem 3.8:Alternate Interior Angles Converse • If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. • Again, you must know that alternate interior angles are congruent.

  32. 1 2 3 4 5 6 8 7 Theorem 3.9:Consecutive Interior Angles Converse • If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. • Be sure that the consecutive interior angles are supplementary. 180o = + + = 180o

  33. 1 2 4 3 8 7 5 6 Theorem 3.10:Alternate Exterior Angles Converse • If two lines are cut by a transversal so that alternate exterior angles are congruent, then lines are parallel. • Again be sure that the alternate exterior angles are congruent.

  34. Example 4 Yes they are parallel! Yes they are parallel! Is it possible to prove the lines are parallel? If so, explain how. Because Alternate Interior Angles are congruent. Because Corresponding Angles are congruent. Corresponding Angles Converse Alternate Interior Angles Converse Yes they are parallel! Because Alternate Exterior Angles are congruent. No they are not parallel! Alternate Exterior Angles Converse No relationship between those two angles.

  35. Example 5 Find the value of x that makes the mn. x = 2x – 95 (AIA)‏ 100 = 4x – 28 (CA)‏ (3x + 15) + 75 = 180(CIA)‏ -x = –95 (SPOE)‏ 128 = 4x (APOE)‏ 3x + 90 = 180 (CLT)‏ x = 95 (DPOE)‏ x = 32 (DPOE)‏ 3x = 90 (SPOE)‏ x = 30 (DPOE)‏ Directions do not ask for reasons, I am showing you them because I am a teacher!!

  36. Homework 3.4 • In Class • 1, 3-9 • p153-156 • Homework • 10-35, 37, 38 • Due Tomorrow

  37. Lesson 3.5 Using Properties of Parallel Lines

  38. Lesson 3.5 Objectives • Prove more than two lines are parallel to each other. • Identify all possible parallel lines in a figure.

  39. q p r Theorem 3.11:3 Parallel Lines Theorem • If two lines are parallel to the same line, then they are parallel to each other. • This looks like the transitive property for parallel lines. If p // q and q // r, then p // r.

  40. m n p Theorem 3.12:Parallel Perpendicular Lines Theorem • In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If m  p and n  p, then m // n.

  41. b a c x 125o y 55o z Finding Parallel Lines • Find any lines that are parallel and explain why.

  42. Results • x // y • Corresponding Angles Converse • Postulate 16 • y // z • Consecutive Interior Angels Converse • Theorem 3.9 • x // z • 3 Parallel Lines Theorem • Theorem 3.11 • b // c • Alternate Exterior Angles Converse • Theorem 3.10

  43. Homework 3.5 • In Class • 16, 19 • p160-163 • Homework • 8-24, 33-36, 43-51 • Due Tomorrow

  44. Lesson 3.6 Parallel Lines in the Coordinate Plane

  45. Lesson 3.6 Objectives • Review the slope of a line • Identify parallel lines based on their slopes • Write equations of parallel lines in a coordinate plane

  46. Slope • Recall that slope of a nonvertical line is a ratio of the vertical change divided by the horizontal change. • It is a measure of how steep a line is. • The larger the slope, the steeper the line is. • Slope can be negative or positive whether or not the lines slants up or down. rise Remember that slope is often referred to as run y2 – y1 = m Which really means x2 – x1 y = mx + b Which we find it in an equation by looking for m.

  47. A( , ) B( , ) y2 – y1 = = x2 – x1 Example of Slope You are given two points: 1 2 3 8 Now label each point as 1 and 2. 1 2 Then substitute as the formula for slope tells you. – 8 2 6 3 = – 3 1 2

  48. Postulate 17: Slopes of Parallel Lines Postulate • In a coordinate plane, two nonvertical lines are parallel if and only if they have the sameslope. • Any two vertical lines are parallel. m = -1 m = undefined m = -1

  49. Writing an Equation inSlope Intercept Form • You will be given • Slope • Or at least two points so you can calculate slope • y-intercept Your final answer should always appear in this form. y = mx + b y-intercept slope This is the point at which the line touches the y-axis.

  50. Writing an Equation GivenSlope and 1 Point • For this, you will be given the slope • Or have to determine it from and equation • Or determine it from a set of two points • You will also be given 1 point through which the line passes • To solve, use your slope-intercept form to find b • y = mx+b • Plug in • Slope for m. • The x-value from your point for x. • The y-value from your point for y. • Solve for b using algebra • When finished, be sure to rewrite in slope intercept form using your new m and b. • Leave x and y as x and y in your final equation.

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