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Geometry Section 2-7

J
Jimbo Lamb

This document defines key terms and concepts related to parallel lines and transversals, including: - A transversal is a line that intersects two other lines at two different points. - Interior angles are formed between the two lines, while exterior angles are outside the lines. - Alternate interior angles, corresponding angles, and other angle relationships are defined. - Examples demonstrate identifying angle relationships and using properties of parallel lines to calculate unknown angle measures.

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SECTION 2-7 Parallel Lines and Transversals
ESSENTIAL QUESTIONS How do you identify the relationships between two lines or two planes? How do you name angle pairs formed by parallel lines and transversals?
VOCABULARY 1. Transversal: 2. Interior Angles: 3. Exterior Angles: 4. Consecutive Interior Angles:
VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: 3. Exterior Angles: 4. Consecutive Interior Angles:
VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: Angles that are formed in the region between two lines being intersected by a transversal 3. Exterior Angles: 4. Consecutive Interior Angles:
VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: Angles that are formed in the region between two lines being intersected by a transversal 3. Exterior Angles: Angles that are formed in the region outside of the two lines being intersected by a transversal 4. Consecutive Interior Angles:
VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: Angles that are formed in the region between two lines being intersected by a transversal 3. Exterior Angles: Angles that are formed in the region outside of the two lines being intersected by a transversal 4. Consecutive Interior Angles: Interior angles that are on the same side of the transversal, thus being “next” to each other
VOCABULARY 5. Alternate Interior Angles: 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines:
VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines:
VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines: Nonadjacent exterior angles that are on opposite sides of the transversal
VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines: Nonadjacent exterior angles that are on opposite sides of the transversal Angles that are on the same side of the transversal and on the same side of the lines being intersected by the transversal; these angles have the same position
VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines: Two or more lines in the same plane that do not intersect Nonadjacent exterior angles that are on opposite sides of the transversal Angles that are on the same side of the transversal and on the same side of the lines being intersected by the transversal; these angles have the same position
VOCABULARY 9. Skew Lines: 10. Parallel Planes:
VOCABULARY 9. Skew Lines: Two or more lines that are in different planes and do not intersect 10. Parallel Planes:
VOCABULARY 9. Skew Lines: Two or more lines that are in different planes and do not intersect 10. Parallel Planes: Two or more planes that do not intersect
EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC b. A segment that is skew to EH c. A plane that is parallel to plane ABG
EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC FG, AD, EH b. A segment that is skew to EH c. A plane that is parallel to plane ABG
EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC FG, AD, EH b. A segment that is skew to EH DC, CF, AB, or BG c. A plane that is parallel to plane ABG
EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC FG, AD, EH b. A segment that is skew to EH DC, CF, AB, or BG c. A plane that is parallel to plane ABG Plane DEC
EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. a. ∠2 and ∠6 b. ∠1 and ∠7
EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. a. ∠2 and ∠6 b. ∠1 and ∠7 Corresponding Angles
EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. a. ∠2 and ∠6 b. ∠1 and ∠7 Corresponding Angles Alternate Exterior Angles
EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. c. ∠3 and ∠8 d. ∠3 and ∠5
EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. c. ∠3 and ∠8 d. ∠3 and ∠5 Consecutive Interior Angles
EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. c. ∠3 and ∠8 d. ∠3 and ∠5 Consecutive Interior Angles Alternate Interior Angles
EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. a. ∠1 and ∠2 b. ∠2 and ∠3
EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. a. ∠1 and ∠2 b. ∠2 and ∠3 v; Corresponding
EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. a. ∠1 and ∠2 b. ∠2 and ∠3 v; Corresponding v; Alternate Interior
EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. c. ∠4 and ∠5
EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. c. ∠4 and ∠5 y; Consecutive Interior
POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE
POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CORRESPONDING ANGLES IS CONGRUENT.
POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CORRESPONDING ANGLES IS CONGRUENT. ∠1 ≅ ∠5
POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM
POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE INTERIOR ANGLES IS CONGRUENT.
POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE INTERIOR ANGLES IS CONGRUENT. ∠4 ≅ ∠6
POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM
POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS SUPPLEMENTARY.
POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS SUPPLEMENTARY. ∠4 AND ∠5 ARE SUPPLEMENTARY
POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM
POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS CONGRUENT.
POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS CONGRUENT. ∠2 ≅ ∠7
EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 b. m∠3 c. m∠6
EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES THM. WITH ∠4 b. m∠3 c. m∠6
EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES THM. WITH ∠4 b. m∠3 129°; SUPPLEMENTARY THM. WITH ∠4 c. m∠6
EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES THM. WITH ∠4 b. m∠3 129°; SUPPLEMENTARY THM. WITH ∠4 c. m∠6 51°; ALTERNATE INTERIOR ANGLES THM. WITH ∠4
EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE.
EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2
EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠2
EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠2 m∠4 = 125°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠3
EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠2 m∠4 = 125°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠3 m∠5 = 55°; SUPPLEMENTARY ANGLES THM. WITH ∠4
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x.
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x +10+10
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x +10+10 x = 25
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x +10+10 x = 25 SINCE THE ANGLES ARE CORRESPONDING, THEY ARE CONGRUENT BY THE CORRESPONDING ANGLES POSTULATE, SO THEIR MEASURES ARE EQUA BY THE DEF. OF CONGRUENT ANGLES.
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y.
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180 8y ­ 100 = 180
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180 8y ­ 100 = 180 8y = 280
EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180 8y ­ 100 = 180 8y = 280 y = 35

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Geometry Section 2-7

  • 1. SECTION 2-7 Parallel Lines and Transversals
  • 2. ESSENTIAL QUESTIONS How do you identify the relationships between two lines or two planes? How do you name angle pairs formed by parallel lines and transversals?
  • 3. VOCABULARY 1. Transversal: 2. Interior Angles: 3. Exterior Angles: 4. Consecutive Interior Angles:
  • 4. VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: 3. Exterior Angles: 4. Consecutive Interior Angles:
  • 5. VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: Angles that are formed in the region between two lines being intersected by a transversal 3. Exterior Angles: 4. Consecutive Interior Angles:
  • 6. VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: Angles that are formed in the region between two lines being intersected by a transversal 3. Exterior Angles: Angles that are formed in the region outside of the two lines being intersected by a transversal 4. Consecutive Interior Angles:
  • 7. VOCABULARY 1. Transversal: A line that intersects two other coplanar lines at two different points 2. Interior Angles: Angles that are formed in the region between two lines being intersected by a transversal 3. Exterior Angles: Angles that are formed in the region outside of the two lines being intersected by a transversal 4. Consecutive Interior Angles: Interior angles that are on the same side of the transversal, thus being “next” to each other
  • 8. VOCABULARY 5. Alternate Interior Angles: 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines:
  • 9. VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines:
  • 10. VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines: Nonadjacent exterior angles that are on opposite sides of the transversal
  • 11. VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines: Nonadjacent exterior angles that are on opposite sides of the transversal Angles that are on the same side of the transversal and on the same side of the lines being intersected by the transversal; these angles have the same position
  • 12. VOCABULARY 5. Alternate Interior Angles: Nonadjacent interior angles that are on opposite sides of the transversal 6. Alternate Exterior Angles: 7. Corresponding Angles: 8. Parallel Lines: Two or more lines in the same plane that do not intersect Nonadjacent exterior angles that are on opposite sides of the transversal Angles that are on the same side of the transversal and on the same side of the lines being intersected by the transversal; these angles have the same position
  • 13. VOCABULARY 9. Skew Lines: 10. Parallel Planes:
  • 14. VOCABULARY 9. Skew Lines: Two or more lines that are in different planes and do not intersect 10. Parallel Planes:
  • 15. VOCABULARY 9. Skew Lines: Two or more lines that are in different planes and do not intersect 10. Parallel Planes: Two or more planes that do not intersect
  • 16. EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC b. A segment that is skew to EH c. A plane that is parallel to plane ABG
  • 17. EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC FG, AD, EH b. A segment that is skew to EH c. A plane that is parallel to plane ABG
  • 18. EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC FG, AD, EH b. A segment that is skew to EH DC, CF, AB, or BG c. A plane that is parallel to plane ABG
  • 19. EXAMPLE 1 Identify each of the following using the box. a. All segments parallel to BC FG, AD, EH b. A segment that is skew to EH DC, CF, AB, or BG c. A plane that is parallel to plane ABG Plane DEC
  • 20. EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. a. ∠2 and ∠6 b. ∠1 and ∠7
  • 21. EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. a. ∠2 and ∠6 b. ∠1 and ∠7 Corresponding Angles
  • 22. EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. a. ∠2 and ∠6 b. ∠1 and ∠7 Corresponding Angles Alternate Exterior Angles
  • 23. EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. c. ∠3 and ∠8 d. ∠3 and ∠5
  • 24. EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. c. ∠3 and ∠8 d. ∠3 and ∠5 Consecutive Interior Angles
  • 25. EXAMPLE 2 Refer to the figure below. Classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. c. ∠3 and ∠8 d. ∠3 and ∠5 Consecutive Interior Angles Alternate Interior Angles
  • 26. EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. a. ∠1 and ∠2 b. ∠2 and ∠3
  • 27. EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. a. ∠1 and ∠2 b. ∠2 and ∠3 v; Corresponding
  • 28. EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. a. ∠1 and ∠2 b. ∠2 and ∠3 v; Corresponding v; Alternate Interior
  • 29. EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. c. ∠4 and ∠5
  • 30. EXAMPLE 3 The driveways at a bus station are shown. Identify the transversal connecting each pair of angles in the figure. Then classify the relationship between each pair of angles. c. ∠4 and ∠5 y; Consecutive Interior
  • 32. POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CORRESPONDING ANGLES IS CONGRUENT.
  • 33. POSTULATES AND THEOREMS CORRESPONDING ANGLES POSTULATE IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CORRESPONDING ANGLES IS CONGRUENT. ∠1 ≅ ∠5
  • 34. POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM
  • 35. POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE INTERIOR ANGLES IS CONGRUENT.
  • 36. POSTULATES AND THEOREMS ALTERNATE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE INTERIOR ANGLES IS CONGRUENT. ∠4 ≅ ∠6
  • 37. POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM
  • 38. POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS SUPPLEMENTARY.
  • 39. POSTULATES AND THEOREMS CONSECUTIVE INTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF CONSECUTIVE INTERIOR ANGLES IS SUPPLEMENTARY. ∠4 AND ∠5 ARE SUPPLEMENTARY
  • 40. POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM
  • 41. POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS CONGRUENT.
  • 42. POSTULATES AND THEOREMS ALTERNATE EXTERIOR ANGLES THEOREM IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL, THEN EACH PAIR OF ALTERNATE EXTERIOR ANGLES IS CONGRUENT. ∠2 ≅ ∠7
  • 43. EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 b. m∠3 c. m∠6
  • 44. EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES THM. WITH ∠4 b. m∠3 c. m∠6
  • 45. EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES THM. WITH ∠4 b. m∠3 129°; SUPPLEMENTARY THM. WITH ∠4 c. m∠6
  • 46. EXAMPLE 4 IN THE FIGURE, m∠4 = 51°. FIND THE MEASURE OF EACH ANGLE. GIVE A JUSTIFICATION TO YOUR ANSWER. a. m∠2 51°; VERTICAL ANGLES THM. WITH ∠4 b. m∠3 129°; SUPPLEMENTARY THM. WITH ∠4 c. m∠6 51°; ALTERNATE INTERIOR ANGLES THM. WITH ∠4
  • 47. EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE.
  • 48. EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2
  • 49. EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠2
  • 50. EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠2 m∠4 = 125°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠3
  • 51. EXAMPLE 5 USE THE FIGURE, IN WHICH a||b, m∠2 = 125°, AND c||d||e, TO FIND THE MEASURE OF EACH NUMBERED ANGLE. PROVIDE A REASON FOR THE ANSWER FOR EACH MEASURE. m∠1 = 125°; VERTICAL ANGLES THM. WITH ∠2 m∠3 = 55°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠2 m∠4 = 125°; CONSECUTIVE INTERIOR ANGLES THM. WITH ∠3 m∠5 = 55°; SUPPLEMENTARY ANGLES THM. WITH ∠4
  • 52. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x.
  • 53. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15
  • 54. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x
  • 55. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x +10+10
  • 56. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x +10+10 x = 25
  • 57. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. a. IF m∠2 = (2x ­ 10)° AND m∠6 = (x + 15)°, FIND x. 2x ­ 10 = x + 15 ­x ­x +10+10 x = 25 SINCE THE ANGLES ARE CORRESPONDING, THEY ARE CONGRUENT BY THE CORRESPONDING ANGLES POSTULATE, SO THEIR MEASURES ARE EQUA BY THE DEF. OF CONGRUENT ANGLES.
  • 58. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y.
  • 59. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180
  • 60. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180
  • 61. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180 8y ­ 100 = 180
  • 62. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180 8y ­ 100 = 180 8y = 280
  • 63. EXAMPLE 6 USE THE FIGURE TO FIND THE INDICATED VARIABLE. EXPLAIN YOUR REASONING. b. IF m∠7 = [4(y ­ 25)]° AND m∠1 = (4y)°, FIND y. 4(y ­ 25) + 4y = 180 4y ­ 100 + 4y = 180 8y ­ 100 = 180 8y = 280 y = 35