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3.1 Lines and Angles

3.1 Lines and Angles

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3.1 Lines and Angles

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  1. 3.1 Lines and Angles Day 1 Part 1 CA Standards 7.0, 16.0

  2. Warmup • Define the following terms. • Line • Perpendicular lines • Parallel lines • Skew lines Two lines that intersect and form 90° Two lines that are coplanar and do not intersect Two lines that do not intersect and are not coplanar

  3. Parallel Postulate • If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P . l There is exactly one line through P parallel to l.

  4. 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. . P l There is exactly one line through P perpendicular to l.

  5. A transversal is a line that intersects two or more coplanar lines at different points. Alternate exterior angles 1 1 2 2 Corresponding angles 4 4 3 3 Consecutive interior angles 5 5 6 6 6 8 8 7 Alternate interior angles

  6. List all pairs of angles that are corresponding angles. 2 7 8 4 1 3 5 6

  7. Try it again…. • List all pairs of corresponding alternate exterior, alternate interior and consecutive interior angles. 1 2 8 7 3 4 6 5 9 10 11 12 16 15 14 13

  8. Challenge 1 3 5 6 7 2 4 8 10 11 12 9 Name all pairs of corresponding angles, alternate exterior angles and consecutive interior angles.

  9. 3.2 Proof and Perpendicular Lines Day 1 Part 2 CA Standards 2.0, 4.0

  10. Types of Proofs • 1. Two column proof: It is the most formal type of proof. It lists numbered statements in the left column and a reason for each statement in the right column. • 2. Paragraph proof: It is a type of proof describes the logical argument with sentences. It is more conversational than a two-column proof. • 3. Flow proof: It is a type of proof uses the same statements and reasons as a two-column proof, but the logical flow connecting the statements is indicated by arrows.

  11. Theorems • If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. g ( ( h line g is perpendicular to line h

  12. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. L

  13. Find the value of x x ° 50 °

  14. If two lines are perpendicular, then they intersect to form four right angles.

  15. Find the value of k k 40 °

  16. T • Complete the statement given that s t. • If m<1 = 38 °, then m<4=____ • m<2 = _____ • If m<6 = 51 °, then m<1=____ • If m<3 = 42 °, then m<1 = ____ 6 5 1 4 2 3

  17. Proof… • Given: x + y = 60, x = 5 Prove: y = 55 • Given: m<5 = m<6, <5 and <6 are a linear pair. Prove: j ┴ k

  18. Pg. 132 # 10 - 31 • Pg. 139 # 18 – 19 • Pg. 141 # 29 – 36

  19. Ch 3.3 Parallel Lines and Transversals Day 2 Part 1 CA Standards 7.0

  20. Warmup • List all pairs of angles that fit the description. • Corresponding • Alternate exterior • Alternate interior • Consecutive interior 1 8 2 7 3 6 4 5

  21. Corresponding Angles Postulate • If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 > 2 >

  22. Find the value of x 2x + 10 < <

  23. Alternate Interior Angles Theorem • If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. > 3 4 >

  24. Consecutive Interior Angles Theorem • If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. > 5 6 >

  25. Find the value of the variable. < 4x x <

  26. Alternate Exterior Angles Theorem • If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 > > 8

  27. Solve for u, given r // s 90 – u r s u

  28. Find the values of x and y x > 67 > y

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

  30. Find the value of the variables. 2w – 10 < 70° < g < 130° <

  31. Challenge…Solve for the variables > > 4x 6x + y x + 5y

  32. 3.4 Proving Lines are Parallel Day 2 Part 2 CA Standards 4.0, 7.0

  33. Review • State the postulate or theorem that justifies the statement. a b > c d f e > g h

  34. What is the converse of the following statement? • If , then n // m. • A. if and only if n // m. • B. If , then n // m. • C. If n // m, then • D. only if n // m.

  35. Write a converse of the following statement • If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. • If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

  36. Write a converse of the following… • Corresponding Angles Postulate • Consecutive Interior Angles Theorem • Alternate Exterior Angles Theorem

  37. Is it possible to prove that lines a and b are parallel? 120 60 b a

  38. Find the value of the variables (if possible.) k 125° h

  39. What value of x would make lines l1 and l2 parallel? l1 2x + 4 3x - 9 l2

  40. Find the value of the variable. < (2w + 4) (3w – 9) <

  41. Review • Complete the statement. • If m<1 = 23°, then m<2=___ • If m<4 = 69°, then m<3=___ • If m<2 = 70°, then m<4=___ 3 2 1 4 L

  42. Review • Given: <1 <2 • Prove: m ll n 3 m 2 1 n

  43. Review • Given: <4 and <5 are supplementary. • Prove: g ll h g 5 4 h

  44. Which lines, if any, are parallel? 69 80 32 31 m n j k

  45. Use the diagram to complete the statement. <6 <12 and ___ : alt. ext. <10 and ___ : corresp. <10 and ___ : alt. int. < 9 and ___ : cons. Int. <6 5 6 <8 8 7 <8 10 9 12 11

  46. Pg. 146 # 8 – 27 • Pg. 153 # 3 – 28, 34

  47. Ch 3.5 Using Properties of Parallel Lines Day 3 Part 1 CA Standards 7.0, 16.0

  48. P.O.D. • Define and give example. • 1. corresponding angles • 2. alternate interior angles • 3. alternate exterior angles • 4. consecutive interior angles • 5. vertical angles

  49. Given: m ||n, n ||kProve: m ||k 1 m 2 n 3 k

  50. Theorem • If two lines are parallel to the same line, then they are parallel to each other. r p q If p // q and q // r, then p // r.