September 30, 2016 3.1 Pairs of Lines and Angles 30, 2016 3.1 Pairs of Lines and Angles Parallel...

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?



Parallel Lines

Coplanar lines that do not

intersect.

m n

m || n

Small arrows are used in a diagram

to show lines are parallel.


Skew Lines

Lines that do not intersect and are not

coplanar.

r

s


Parallel PlanesPlanes that don’t intersect.


Segments and Rays can be

parallel.

CDAB ||

OPMN ||

Sketch the following examples.

P

A

B

C

D

O

M

N


Visualization

A

B D

E

F

G

Think of a

rectangular box.

ED and ABParallel

EF and ABSkew

BD and ABPerpendicular


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.


Your turnL

N

M

S

Q

R

P

Plane SNM


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.


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


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!


TransversalsA transversal cuts across two parallel

lines at an angle.

The transversal intersects the two lines

at two different points.


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.


This means ALL corresponding

angles are congruent.

1 2

34

5 6

78

1 5

2 6

3 7

4 8


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


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


Theorem 3.3 Alternate Exterior

Angles Theorem


transversal, then alternate exterior angles

are congruent.

2 lines || alt ext s


Theorem 3.4 Same Side Interior

Angles Theorem


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


Theorems in a nutshell.

2 lines || corr s



2 lines || alt. int. s



2 lines || alt. ext. s



2 lines || ss int. s supp.


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°


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.


a. b.

x° y°

70°x°

y°

120°


Example 4

m

n

m || n

(120 – x)°

5x°

Solve for x.

2 lines || alt ext s

5x = 120 – x

6x = 120

x = 20


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


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)°


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.


a.b.

x°y°

100°

(2x - 10)°

80°

Extra Practice

In each of the following problems solve for

x and y.


c. d.

x°

y°

130°

3x°

120°

Extra PracticeSolve for x and y.


5x = 35

Extra PracticeIn each of the following problems solve for

x and y.


e.


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.)


Solution

m

n

m || nx°43°

25°

Find x.

25°

43°

x° = 25° + 43° = 68°

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