parallel lines & transversals 3.3. transversal a line, ray, or segment that intersects 2 or more...
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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