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Analytic Geometry January 14, 2019

Proving Theorems About Angles in Parallel Lines Cut by Transversals

EQ: How do angle relationships work together in a set of parallel lines intersected by a transversal?

MGSE9­12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

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Warm­up:

x2 + 10xo

4x + 16o

Find the value(s) of x.

x = _________

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Homework Answers1. 2. 3. 4.

5. 6. 7. 8.

109˚ 71˚ 109˚ 71˚

71˚ 109˚ 71˚ 115˚

9. x = 25˚ = = 100˚ 10. x = 6˚ = 102˚

11. 60˚ 12. 4˚ or ‐3˚ 13. x = y =

14. x = y =

8˚ 12˚

13˚ 47˚

alt. ext. <'s ≅ linear pr. sum to 180o vertical <'s are ≅ alt. int. <'s ≅

vertical <'s are ≅ vertical <'s are ≅ vertical <'s are ≅ vertical <'s are ≅

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Homework Answers

alt. ext. <'s ≅linear pr. sum to 180o vertical <'s are ≅

alt. int. <'s ≅vertical <'s are ≅ vertical <'s are ≅

vertical <'s are ≅

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60o

82o1

1

2

34o

42o

3

1

40o

15o2

80o

3

45

6

7 89

Applications of Parallel LinesAngle relationships can be used to find missing angle measures in geometric figures and diagrams.

Examples:Find the indicated angle measures.

1. m∠1 = ________ m∠2 = ________

2. m∠1 = ________

3. m∠1 = ________ 4. m∠1 = ________m∠2 = ________m∠3 = ________

5. m∠1 = ________; m∠2 = ________

m∠3 = ________; m∠4 = ________

m∠5 = ________; m∠6 = ________

m∠7 = ________; m∠8 = ________

m∠9 = ________

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60o

82o1

1

2

34o

42o

3

1

40o

15o2

80o

3

45

6

7 89

Applications of Parallel LinesAngle relationships can be used to find missing angle measures in geometric figures and diagrams.

Examples:Find the indicated angle measures.

1. m∠1 = ________ m∠2 = ________

2. m∠1 = ________

3. m∠1 = ________ 4. m∠1 = ________m∠2 = ________m∠3 = ________

5. m∠1 = ________; m∠2 = ________

m∠3 = ________; m∠4 = ________

m∠5 = ________; m∠6 = ________

m∠7 = ________; m∠8 = ________

m∠9 = ________

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105o

57o

4x+14o

6y+38o

3y+83o

A

CD

76o

5x+21o

B

Find the values of x or y.6. 7.

8. m∠BAD = 112°

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105o

57o

4x+14o

6y+38o

3y+83o

A

CD

76o

5x+21o

B

Find the values of x or y.6. 7.

8. m∠BAD = 112°

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ProofsFor #9‐10, fill in the blank with the correct reason from the given list.9. Vertical Angles are ≅. Congruent angles have the same measure.

Transitive Property Given Corresponding angles are ≅.Substitution Property

Given: p || q Prove: ∠1 ≅ ∠2

Statements Reasons

1. p || q 1. _______________________________2. ∠1 ≅ ∠3 2. _______________________________3. ∠3 ≅ ∠2 3. _______________________________4. ∠1 ≅∠2 4. _______________________________

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ProofsFor #9‐10, fill in the blank with the correct reason from the given list.9. Vertical Angles are ≅. Congruent angles have the same measure.

Transitive Property Given Corresponding angles are ≅.Substitution Property

Given: p || q Prove: ∠1 ≅ ∠2

Statements Reasons

1. p || q 1. _______________________________2. ∠1 ≅ ∠3 2. _______________________________3. ∠3 ≅ ∠2 3. _______________________________4. ∠1 ≅∠2 4. _______________________________

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10. Linear Pair sum to 180° Alternate Interior Angles are ≅. Given Definition of congruent angles Definition of supplementary angles

Substitution Property

Given: p || q Prove: ∠1 and ∠2 are supplementary.

Statements Reasons

1. p || q 1. _______________________________2. ∠1 ≅ ∠3 2. _______________________________3. m∠1 = m∠3 3. _______________________________4. m∠3 + m∠2 = 180° 4. _______________________________5. m∠1 + m∠2 = 180° 5. _______________________________6. ∠1 and ∠2 are 6. _______________________________

supplementary.

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10. Linear Pair sum to 180° Alternate Interior Angles are ≅. Given Definition of congruent angles Definition of supplementary angles

Substitution Property

Given: p || q Prove: ∠1 and ∠2 are supplementary.

Statements Reasons

1. p || q 1. _______________________________2. ∠1 ≅ ∠3 2. _______________________________3. m∠1 = m∠3 3. _______________________________4. m∠3 + m∠2 = 180° 4. _______________________________5. m∠1 + m∠2 = 180° 5. _______________________________6. ∠1 and ∠2 are 6. _______________________________

supplementary.

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2

11. Show that ∠1 ≅ ∠2.

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2

11. Show that ∠1 ≅ ∠2.

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Assignment:worksheet

On‐line and textbook help references: pp. 67, 72 ‐ 76 ­ http://www.mathsisfun.com/geometry/parallel‐lines.html­ http://www.regentsprep.org/Regents/math/geometry/GP8/Lparallel.htm­ http://www.khanacademy.org/math/basic‐geo/basic‐geo‐angles/basic‐geo‐angle‐relationships/v/angles‐formed‐by‐parallel‐lines‐and‐transversals

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