september 30, 2016 3.1 pairs of lines and angles 30, 2016 3.1 pairs of lines and angles parallel...
TRANSCRIPT
September 30, 2016 3.1 Pairs of Lines and Angles
Geometry
3.1 Pairs of Lines & Angles
3.2 Parallel Lines and Transversals
Essential Question
What does it mean when two line are
parallel, intersecting, coincident, skew, or
perpendicular?
And what are the properties of angles
formed by parallel lines cut with a
transversal?
September 30, 2016 3.1 Pairs of Lines and Angles
September 30, 2016 3.1 Pairs of Lines and Angles
Parallel Lines
Coplanar lines that do not
intersect.
m n
m || n
Small arrows are used in a diagram
to show lines are parallel.
September 30, 2016 3.1 Pairs of Lines and Angles
Skew Lines
Lines that do not intersect and are not
coplanar.
r
s
September 30, 2016 3.1 Pairs of Lines and Angles
Parallel PlanesPlanes that don’t intersect.
September 30, 2016 3.1 Pairs of Lines and Angles
Segments and Rays can be
parallel.
CDAB ||
OPMN ||
Sketch the following examples.
P
A
B
C
D
O
M
N
September 30, 2016 3.1 Pairs of Lines and Angles
Visualization
A
B D
E
F
G
Think of a
rectangular box.
ED and ABParallel
EF and ABSkew
BD and ABPerpendicular
September 30, 2016 3.1 Pairs of Lines and Angles
Example 1Think of each segment in the figure are part of a
line. Identify each pair of lines as parallel, skew
or perpendicular.
Parallel
Perpendicular
Perpendicular
Skew
AB
D
E F
G
C
Name a ...
Line parallel to
Line perpendicular to
Line skew to
Plane parallel to plane RPL.
September 30, 2016 3.1 Pairs of Lines and Angles
Your turnL
N
M
S
Q
R
P
Plane SNM
September 30, 2016 3.1 Pairs of Lines and Angles
Postulate 3.1 Parallel Postulate
If there is a line and a point not on
the line, then there is exactly one
line through the point parallel to
the given line.
September 30, 2016 3.1 Pairs of Lines and Angles
Postulate 3.2 Perpendicular PostulateIf there is a line and a point not on
the line, then there is exactly one
line through the point
perpendicular to the given line.
Example 2
September 30, 2016 3.1 Pairs of Lines and Angles
The given line markings show
how the roads in a town are
related to one another.
Name a pair of parallel lines.
Name a pair of perpendicular
lines.
Is
𝐷𝑀 𝑎𝑛𝑑 𝐹𝐸
𝐷𝑀 𝑎𝑛𝑑 𝐵𝐹
No!
September 30, 2016 3.1 Pairs of Lines and Angles
TransversalsA transversal cuts across two parallel
lines at an angle.
The transversal intersects the two lines
at two different points.
September 30, 2016 3.1 Pairs of Lines and Angles
This is not a transversal.
The lines intersect at
only one point.
September 30, 2016 3.2 Parallel Lines and Transversals
Theorem 3.1:Corresponding Angles
m
n
m || n
If two parallel lines are cut by a transversal, then the
pairs of corresponding angles are congruent.
September 30, 2016 3.2 Parallel Lines and Transversals
This means ALL corresponding
angles are congruent.
1 2
34
5 6
78
1 5
2 6
3 7
4 8
September 30, 2016 3.2 Parallel Lines and Transversals
Example 1Find all angle measures in the picture.
60°
?60°
?60°
?60°
?120°
?120°
?120°
?120°
Notice…When two parallel lines are cut by a transversal, any pair of angles will either be ____________________ or _____________________.
congruentsupplementary
September 30, 2016 3.2 Parallel Lines and Transversals
Theorem 3.2 Alternate Interior Angles
Theorem
If two parallel lines are cut by a
transversal, then alternate interior angles
are congruent.
2 lines || alt int s
September 30, 2016 3.2 Parallel Lines and Transversals
Theorem 3.3 Alternate Exterior
Angles Theorem
If two parallel lines are cut by a
transversal, then alternate exterior angles
are congruent.
2 lines || alt ext s
September 30, 2016 3.2 Parallel Lines and Transversals
Theorem 3.4 Same Side Interior
Angles Theorem
If two parallel lines are cut by a
transversal, then the pairs of same side
interior angles are supplementary.
2 lines || ss int s supp
1
2
3
4
m1 + m2 = 180
m3 + m4 = 180
September 30, 2016 3.2 Parallel Lines and Transversals
Theorems in a nutshell.
2 lines || corr s
September 30, 2016 3.2 Parallel Lines and Transversals
Theorems in a nutshell.
2 lines || alt. int. s
September 30, 2016 3.2 Parallel Lines and Transversals
Theorems in a nutshell.
2 lines || alt. ext. s
September 30, 2016 3.2 Parallel Lines and Transversals
Theorems in a nutshell.
2 lines || ss int. s supp.
September 30, 2016 3.2 Parallel Lines and Transversals
2 lines ||
corrs
alt int s
alt ext s
ss int s supp
These are the “reasons” for proof.
Example 2
State the theorem that justifies the
statement below.
3 6
4 5
1 4
m6 + m7 = 180°
September 30, 2016 3.2 Parallel Lines and Transversals
Corr. s
Alt Int s
Alt Ext s
Same Side Int s
Example 3
Solve each problem for x and y. Identify
the theorem that justifies your answer.
September 30, 2016 3.2 Parallel Lines and Transversals
a. b.
x° y°
70°x°
y°
120°
September 30, 2016 3.2 Parallel Lines and Transversals
Example 4
m
n
m || n
(120 – x)°
5x°
Solve for x.
2 lines || alt ext s
5x = 120 – x
6x = 120
x = 20
September 30, 2016 3.2 Parallel Lines and Transversals
Example 5
m
n
m || n(x + 20)°
(x + 8)°
Solve for x.
2 lines || SS int s supp
(x + 20) + (x + 8) = 180
2x + 28 = 180
2x = 152
x = 76
September 30, 2016 3.2 Parallel Lines and Transversals
Example 6
m
n
m || n
(x + 40)°
(x + 50)°
Solve for x.
Linear Pair Post.
(x + 40) + (x + 50) = 180
2x + 90 = 180
2x = 90
x = 45
2 lines || corr s
(x + 40)°
September 30, 2016 3.2 Parallel Lines and Transversals
In Summary.
2 lines ||
corr s
alt int s
alt ext s
ss int s supp
Extra Practice
In each of the following problems solve for
x and y.
September 30, 2016 3.2 Parallel Lines and Transversals
a.b.
x°y°
100°
(2x - 10)°
80°
Extra Practice
In each of the following problems solve for
x and y.
September 30, 2016 3.2 Parallel Lines and Transversals
c. d.
x°
y°
130°
3x°
120°
Extra PracticeSolve for x and y.
September 30, 2016 3.2 Parallel Lines and Transversals
5x = 35
Extra PracticeIn each of the following problems solve for
x and y.
September 30, 2016 3.2 Parallel Lines and Transversals
e.
September 30, 2016 3.2 Parallel Lines and Transversals
Extra for Experts
m
n
m || nx°43°
25°
Find x. (Hint: Draw a line through the vertex of angle x and parallel to the other two lines.)
September 30, 2016 3.2 Parallel Lines and Transversals
Solution
m
n
m || nx°43°
25°
Find x.
25°
43°
x° = 25° + 43° = 68°