3.1 Identify Pairs of Lines and Angles
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3.1 Identify Pairs of Lines and Angles. Objectives. Identify the relationships between two lines or two planes Name angles formed by a pair of lines and a transversal. Relationships Between Lines.

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Objectives
Objectives

  • Identify the relationships between two lines or two planes

  • Name angles formed by a pair of lines and a transversal


Relationships between lines
Relationships Between Lines

  • Parallel Lines– lines that are coplanar and do not intersect; We use the symbol ||to represent parallel lines, (i.e. AB || CD).

  • Skew Lines– lines that do not intersect and are not coplanar

  • Parallel Planes– two planes that do not intersect


EXAMPLE 1

Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure appear to fit the description?

a.

Line(s) parallel to CDand containing point A

b.

Line(s) skew to CDand containing point A

Line(s) perpendicular to CDand containing point A

c.

d.

Plane(s) parallel to plane EFGand containing point A

EXAMPLE 1

Identify Relationships in Space


EXAMPLE 1 (solution)

AB, HG, and EFall appear parallel to CD, but only ABcontains point A.

a.

b.

Both AGand AHappear skew to CDand contain point A.

BC, AD, DE, and FCall appear perpendicular to CD, but only ADcontains pointA.

c.

PlaneABCappears parallel to plane EFGand contains point A.

d.

Identify Relationships in Space


EXAMPLE 2

The given line markings show how the roads are related to one another.

a.

Name a pair of parallel lines.

b.

Name a pair of perpendicular lines.

Is FEAC? Explain.

c.

EXAMPLE 2

Identify Parallel and Perpendicular Lines


b.

MDBF

c.

FEis not parallel to AC, because MDis parallel to FEand by the Parallel Postulate there is exactly one line parallel to FEthrough M.

EXAMPLE 2 (solution)

a.

MDFE

EXAMPLE 2

Identify Parallel and Perpendicular Lines


1.

Look at the diagram in Example 1. Name the lines through point Hthat appear skew to CD.

ANSWER

AH , EH

YOUR TURN

GUIDED PRACTICE


ANSWER

Yes; Since A is not on MD and MD is to BF, the Perpendicular Postulate guarantees that there is exactly one line through a point perpendicular to a line, so AC can not be perpendicular to BF also.

YOUR TURN

GUIDED PRACTICE

2.

In Example 2, can you use the Perpendicular Postulate to show that ACis not perpendicular to BF? Explain why or why not.


Postulates
Postulates

  • Postulate 13 (Parallel Postulate)– There is exactly one line through a point that is parallel to a given line.

  • Postulate 14 (Perpendicular Postulate) – There is exactly one line through a point that is perpendicular to a given line.


Relationships between lines and the angles formed

1

2

3

4

5

6

7

8

Relationships Between Lines and the Angles Formed

  • Transversal- a line that intersects two or more coplanar lines at different points

  • Angles 1 and 5 are corresponding angles

  • Angles 1 and 8 are alternate exterior angles


Relationships between lines and the angles formed1

1

2

3

4

5

6

7

8

Relationships Between Lines and the Angles Formed

  • Angles 3 and 6 are alternate interior angles.

  • Angles 3 and 5 are consecutive interior angles orsame side interior angles


EXAMPLE 3

Identify all pairs of angles of the given type.

d.

b.

a.

Corresponding

Alternate interior

d.

Consecutive interior

Alternate exterior

c.

and

and

and

and

and

and

and

and

and

and

∠ 5

∠ 6

∠ 7

∠ 5

∠ 6

∠ 7

∠ 7

∠ 8

∠ 8

∠ 5

1

4

2

2

4

3

3

2

4

1

a.

b.

c.

EXAMPLE 3

Identify Angle Relationships

SOLUTION


YOUR TURN

1.

ANSWER

corresponding angles

GUIDED PRACTICE

Classify the pair of numbered angles.


YOUR TURN

2.

ANSWER

alternate exterior angles.

GUIDED PRACTICE

Classify the pair of numbered angles.


Assignment
Assignment

  • Geometry: Pg. 150 – 152 #3 – 33, 36


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