Parallel Lines & Transversals 3.3Parallel Lines & Transversals 3.3

Transversal

A line, ray, or segment that intersects 2 or more COPLANAR lines, rays, or segments.

Non-Parallel lines

transversal

Parallel lines

transversal

1

5

Corresponding Angles PostulateCorresponding Angles Postulate

23 4

67 8

1 5 2 6 3 7 4 81 5

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

2 6 3 7 4 8

Alternate Interior Angles PostulateAlternate Interior Angles Postulate12

4

67 8

4 6

3

5

3 53 5 4 6

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Consecutive Interior Angles PostulateConsecutive Interior Angles Postulate12

4

67 8

3

5

m 4 + m 5 = 180°

m 3 + m 6 = 180°

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

m 4 + m 5 = 180°

m 3 + m 6 = 180°

Alternate Exterior Angles PostulateAlternate Exterior Angles Postulate12

4

67 8

3

5

1 71 7 2 8

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

2 8

j k

Perpendicular Transversal TheoremPerpendicular Transversal Theorem

If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.

Prove the Alternate Interior Angles TheoremProve the Alternate Interior Angles Theorem..

GIVEN p || q

Statements Reasons

p || q1

PROVE 1 2

2 1 3

3 3 2

4 1 2

1 Given

2 Corresponding Angles Postulate

3 Vertical Angles Theorem

4 Transitive property of Congruence

Using Properties of Parallel Lines

Given that m5 = 65°, find each measure. Tellwhich postulate or theoremyou use.

Linear Pair Postulate

Alternate Exterior Angles Theorem

Corresponding Angles Postulate

Vertical Angles Theoremm 6 = m 5 = 65°

m 7 = 180° – m 5 = 115°

m 9 = m 7 = 115°

m 8 = m 5 = 65°

m 4 = 125°

m 4 + (x + 15)° = 180°

Use properties ofparallel lines to findthe value of x.

Corresponding Angles Postulate

Linear Pair Postulate

125° + (x + 15)° = 180° Substitute.

PROPERTIES OF SPECIAL PAIRS OF ANGLES

Subtract.x = 40°

Give an example of each angle pair.

A. corresponding angles

B. alternate interior angles

C. alternate exterior angles

1 and 5 or 2 and 6 or 4 and 8 or 3 and 7

D. consecutive interior angles

3 and 5 or 4 and 6

1 and 7 or 2 and 8

3 and 6 or 4 and 5

GIVE AN EXAMPLE OF EACH ANGLE PAIRGIVE AN EXAMPLE OF EACH ANGLE PAIR

A. corresponding angles

B. alternate interior angles

C. alternate exterior angles

1 and 3

D. consecutive interior angles

2 and 7

1 and 8

2 and 3

GIVE AN EXAMPLE OF EACH ANGLE PAIRGIVE AN EXAMPLE OF EACH ANGLE PAIR

Special Angle Relationships

Interior Angles3 & 6 are Alternate Interior angles4 & 5 are Alternate Interior angles3 & 5 are Consecutive Interior angles4 & 6 are Consecutive Interior angles

1

4

2

65

7 8

3

Exterior Angles1 & 8 are Alternate Exterior angles2 & 7 are Alternate Exterior angles1 & 7 are Consecutive Exterior angles2 & 8 are Consecutive Exterior angles

Special Angle RelationshipsWHEN THE LINES ARE PARALLEL

♥Alternate Interior Angles are CONGRUENT

♥Alternate Exterior Angles are CONGRUENT

♥Consecutive Interior Angles are SUPPLEMENTARY

♥ Corresponding Angles are CONGRUENT

♥Consecutive Exterior Angles are SUPPLEMENTARY

14

2

65

7 8

3

If the lines are not parallel, these angle relationships DO NOT EXIST.

Let’s Practice

m1=120°Find all the remaining

angle measures.1

4

2

65

7 8

3

60°

60°

60°

60°

120°

120°

120°

120°

Find the value of x, name the angles.Find the value of x, name the angles.a. b. c.

d. e. f.

g. h. i.

x = 64x = 64 x = 75x = 75 x = 12x = 12

x = 40x = 40 x = 60x = 60 x = 60x = 60

x = 90x = 90 x = 15x = 15 x = 20x = 20

How would you show that the given lines are parallel?

a. a and b b. b and c c. d and f

d. e and g e. a and c

Corresponding

`s Congruent

Consecutive Interior

`s Supplementary

Corresponding

`s Congruent

Calculate the missing

Corresponding

`s Congruent

Consecutive Interior

`s Supplementary

43

Find the value of each variable.

1. x 2. yx = 2 y = 4

Find the value of x and y that make the lines parallel, Find the value of x and y that make the lines parallel, name the angles.name the angles.

a. x b. y 2x + 2 = x + 56

x = 54

Corresponding `s

Congruent

y = 63

2(54) + 2 = 110

110

110

Consecutive Exterior

`s are Supplementary

y + 7 = 70

70

2(63) – 16 = 110

110

IDENTIFY THE TRANSVERSAL, & CLASSIFY EACH ANGLE PAIR

1 2 3 4

5678

9 10 11 12

13141516

p q

r

s

a. 2 and 16

Alternate Exterior ’s

Transversal p

Lines r and s

b. 6 and 7 Transversal r

Lines p and q

Consecutive Interior ’s

A. 1 and 1 and 33

B. 2 and 2 and 66

C. 4 and 4 and 66

transversal transversal ll corresponding corresponding ss

transversal transversal nnalternate interior alternate interior ss

transversal transversal mmalternate exterior alternate exterior ss

IDENTIFY THE TRANSVERSAL, & CLASSIFY EACH ANGLE PAIR

ReviewIf two lines are intersected by a transversal and any of the angle pairs shown below are congruent, then the lines are parallel. This fact is used in the construction of parallel lines.

Assignment

3.3A and 3.3BSection 9 - 33