confidential 1 geometry angles formed by parallel lines and transversals
TRANSCRIPT
CONFIDENTIAL 1
GeometryGeometry
Angles formed by Angles formed by Parallel Lines and Parallel Lines and
TransversalsTransversals
CONFIDENTIAL 2
Warm UpWarm Up
Give an example of each angle pair.
1) Alternate interior angles 2) Alternate exterior angles
3)Same side interior angles
CONFIDENTIAL 3
Parallel, perpendicular and skew linesParallel, perpendicular and skew lines
When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed.
1 23 4
5 67 8
There are several special pairs of angles formed from this figure.
Vertical pairs: Angles 1 and 4 Angles 2 and 3 Angles 5 and 8 Angles 6 and 7
CONFIDENTIAL 4
Supplementary pairs:
Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7
1 23 4
5 67 8
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs.
There are three other special pairs of angles. These pairs are congruent pairs.
CONFIDENTIAL 5
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Corresponding angle postulateCorresponding angle postulate
1 2 3 45 6 7 8
p q
t
1 3 2 4 5 7 6 8
CONFIDENTIAL 6
Using the Corresponding angle postulateUsing the Corresponding angle postulate
Find each angle measure.
800 x0
B
C
A
A) m( ABC)
x = 80 corresponding angles
m( ABC) = 800
CONFIDENTIAL 7
B) m( DEF)
(2x-45)0 = (x+30)0 corresponding angles
m( DEF) = (x+30)0
(2x-45)0
EF
D
(x+30)0
x – 45 = 30 subtract x from both sides
x = 75 add 45 to both sides
= (75+30)0
= 1050
CONFIDENTIAL 8
Now you try!
1) m( DEF)
RSx0
1180
Q
CONFIDENTIAL 9
Remember that postulates are statements that are accepted without proof. Since the
Corresponding Angles postulate is given as a postulate, it can be used to prove the next
three theorems.
CONFIDENTIAL 10
Alternate interior angles theoremAlternate interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent.
1 3 2 4
1 2
4 3
Theorem
Hypothesis Conclusion
CONFIDENTIAL 11
Alternate exterior angles theoremAlternate exterior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate exterior angles are congruent.
5 7 6 8
5 6
8 7
Theorem
Hypothesis Conclusion
CONFIDENTIAL 12
Same-side interior angles theoremSame-side interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Same-side interior angles are supplementary.
m 1 + m 4 =1800 m 2 + m 3 =1800
Theorem
Hypothesis Conclusion
1 2
4 3
CONFIDENTIAL 13
Alternate interior angles theoremAlternate interior angles theorem
1 2
3m
lGiven: l || m
Prove: 2 3
Proof:
1 3l || m
Given Corresponding angles
2 3
2 1
Vertically opposite angles
CONFIDENTIAL 14
A) m( EDF)
Finding Angle measuresFinding Angle measures
1250
B
C
A
x0
D
E F
m( DEF) = 1250
x = 1250
Alternate exterior angles theorem
Find each angle measure.
CONFIDENTIAL 15
B) m( TUS)
13x0 + 23x0 = 1800 Same-side interior angles theorem
m( TUS) = 23(5)0
36x = 180 Combine like terms
x = 5 divide both sides by 36
= 1150
13x0 23x0
U
T
S
R
Substitute 5 for x
CONFIDENTIAL 16
2) Find each angle measure.
Now you try!
B C
ED
(2x+10)0A
(3x-5)0
CONFIDENTIAL 17
A treble string of grand piano are parallel. Viewed from above, the bass strings form transversals to the treble
string. Find x and y in the diagram.
(25x+5y)0
(25x+4y)0
1200
1250
By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250
By the Corresponding Angles Postulates, (25x+4y)0 = 1200
(25x+5y)0 = 1250
- (25x+4y)0 = 1200
y = 5
25x+5(5) = 125
x = 4, y = 5
Subtract the second equation from the first equation
Substitute 5 for y in 25x +5y = 125. Simplify and solve for x.
CONFIDENTIAL 18
3) Find the measure of the acute angles in the diagram.
Now you try!
(25x+5y)0
(25x+4y)0
1200
1250
CONFIDENTIAL 19
Assessment
Find each angle measure:
1270
x0
KJ
L
2) m( BEF)
(7x-14)0
(4x+19)0
G
AABC
FD
H
E
1) m( JKL)
CONFIDENTIAL 20
Find each angle measure:
1
3) m( 1)
(3x+9)0
6x0
A
B
C
D
Y
X
E
Z
4) m( CBY)
CONFIDENTIAL 21
Find each angle measure:
1150
Y0K
M
L
5) m( KLM)
6) m( VYX)
Y
X
W Z
(2a+50)0
V 4a0
CONFIDENTIAL 22
State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures:
12
34
5
7) m 1 = (7x+15)0 , m 2 = (10x-9)0
8) m 3 = (23x+15)0 , m 4 = (14x+21)0
CONFIDENTIAL 23
Parallel, perpendicular and skew linesParallel, perpendicular and skew lines
When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed.
1 23 4
5 67 8
There are several special pairs of angles formed from this figure.
Vertical pairs: Angles 1 and 4 Angles 2 and 3 Angles 5 and 8 Angles 6 and 7
Let’s review
CONFIDENTIAL 24
Supplementary pairs:
Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7
1 23 4
5 67 8
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs.
There are three other special pairs of angles. These pairs are congruent pairs.
CONFIDENTIAL 25
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Corresponding angle postulateCorresponding angle postulate
1 2 3 45 6 7 8
p q
t
1 3 2 4 5 7 6 8
CONFIDENTIAL 26
Using the Corresponding angle postulateUsing the Corresponding angle postulate
Find each angle measure.
800 x0
B
C
A
A) m( ABC)
x = 80 corresponding angles
m( ABC) = 800
CONFIDENTIAL 27
B) m( DEF)
(2x-45)0 = (x+30)0 corresponding angles
m( DEF) = (x+30)0
(2x-45)0
EF
D
(x+30)0
x – 45 = 30 subtract x from both sides
x = 75 add 45 to both sides
= (75+30)0
= 1050
CONFIDENTIAL 28
Alternate interior angles theoremAlternate interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent.
1 3 2 4
1 2
4 3
Theorem
Hypothesis Conclusion
CONFIDENTIAL 29
Alternate exterior angles theoremAlternate exterior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Alternate exterior angles are congruent.
5 7 6 8
5 6
8 7
Theorem
Hypothesis Conclusion
CONFIDENTIAL 30
Same-side interior angles theoremSame-side interior angles theorem
If two parallel lines are cut by a transversal, then the two pairs of Same-side interior angles are supplementary.
m 1 + m 4 =1800 m 2 + m 3 =1800
Theorem
Hypothesis Conclusion
1 2
4 3
CONFIDENTIAL 31
Alternate interior angles theoremAlternate interior angles theorem
1 2
3m
lGiven: l || m
Prove: 2 3
Proof:
1 3l || m
Given Corresponding angles
2 3
2 1
Vertically opposite angles
CONFIDENTIAL 32
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