geometry angles formed by parallel lines and transversals
DESCRIPTION
Geometry Angles formed by Parallel Lines and Transversals. Warm Up. Give an example of each angle pair. 1) Alternate interior angles 2) Alternate exterior angles 3)Same side interior angles. 1 2. 3 4. 5 6. 7 8. Parallel, perpendicular and skew lines. - PowerPoint PPT PresentationTRANSCRIPT
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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