3.1 Lines and Angles

Mrs. SpitzFall 2004

Standard/Objectives:

Standard 3: Students will have a foundation in geometric concepts.

Objectives:• Identify relationships between lines.• Identify angles formed by

transversals.

Homework Assignment

• Pp. 132-133 #1-31

Definitions

• Parallel lines – Two lines are parallel lines if they are coplanar and do not intersect.

• Skew lines—Lines that do not intersect and are not coplanar.

• Parallel planes—two planes that do not intersect.

Example 1: Identifying relationships in space

• Think of each segment in the diagram. Which appear to fit the description?

a. Parallel to AB and contains D

b. Perpendicular to AB and contains D

c. Skew to AB and contains D

d. Name the plane(s) that contains D and appear to be parallel to plane ABE

A

B

D

C

F

E H

G

Solution

a. CD, GH and EF are all parallel to AB, but only CD passes through D and is parallel to AB.

b. BC, AD, AE and BF are all perpendicular to AB, but only AD passes through D and is perpendicular to AB

c. DG, DH, and DE all pass through D and are skew to AB.

d. Only plane DCH contains D and is parallel to plane ABE

Postulate 13: 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.

P

l

Postulate 14: Perpendicular Postulate

• If there is a line and a point not on the line, then there is exactly one line through the given point perpendicular to the given line.

l

P

Perpendicular Construction—pg. 130

• Use the following steps to construct a line that passes through a given point P and is perpendicular to a given line l.

1. Place the compass at point P and draw an arc that intersects the line l twice. Label the intersections A and B.

2. Draw an art with center A. Using the same radius, draw an arc with center B. Label the intersection of the arcs Q.

3. Use a straight edge to draw PQ. PQ l.

Definitions:

• 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

• Angles 3 and 5 are alternate interior angles.

• 3 and 5 are consecutive interior angles

4

7

1 2

35 6

8

Example 2: Identifying Angle relationships

• List all pairs of angles that fit the description.

a. Correspondingb. Alternate exteriorc. Alternate interiord. Consecutive

interior

1

2

3

4

7

6

58

Solution:

a. 1 and 5; 2 and 6; 3 and 7, 4 and 8

b. 1 and 8, 2 and 7c. 3 and 6, 4 and 5d. 3 and 5, 4 and 6