Learn to identify and describe geometric figures.

Course 2

7-1 Points, Lines, and Planes

Learn to identify angles and parts of angles.

Learn to identify parallel, perpendicular, and skew lines, and angles formed by a transversal.

Course 2

7-1 Points, Lines, and Planes

A point is an exact location in space. It is usually represented as a dot, but it has no size at all

• A point AUse a capital letter to namea point.

A line is a straight path that extends without end in opposite directions.

A number line is an example of a line.

Helpful Hint

XY, or YXUse two pointson the line to name a line.

X Y

Course 2

7-1 Points, Lines, and Planes

A ray is a part of a line.It has one endpoint andextends without end inone direction.

GHName the endpointfirst when naminga ray.

A line segment is part of a line. or a raythat extends from oneendpoint to another.

LM, or MLUse the endpointsto name a line segment.

L M

HG

Course 2

7-1 Points, Lines, and Planes

A plane is a perfectly flatsurface that extends infinitelyin all directions.

plane QRSUse three pointsin any order, noton the same line,to name a plane.

A coordinate plane is an example of a plane.

Helpful Hint

Q

R

S

Identify the figures in the diagram.

Additional Example 1: Identifying Points, Lines, and Planes

Course 2

7-1 Points, Lines, and Planes

DE

F

A. three points

B. two lines

C. a plane

D, E, and F

DE, DF

plane DEF

Choose any two points on a line to name the line.Choose any three points, not on the same line, in any order.

Identify the figures in the diagram.

Course 2

7-1 Points, Lines, and Planes

C

AB

D

Try This: Example 2

A. three rays

B. three line segments

BC, CA, BD

BA, CA, BD

Name the endpoint ofa ray first.

Use the endpoints in any order to name a segment.

Course 2

7-1 Points, Lines, and Planes

Figures are congruent if they have the same shape andsize. If you place one on top of the other, they matchexactly. Line segments are congruent if they have thesame length.

Tick marks are used to indicate congruent line segments. In the illustration below, segments that have the same number of tick marks are congruent.Line segments AB and BC are congruent (one tick mark), and line segments MN and OP are congruent(two tick marks).

M P

ON 5 ft

3 ft3 ft

8 ftA C

B

20 m20 m

16 m

Identify the line segments that are congruent.

Additional Example 3: Identifying Congruent Line Segments

Course 2

7-1 Points, Lines, and Planes

A B

DC

E FAB CD

AC BD

BF DF EC AE

One tick mark

Two tick marks

Three tick marks

The symbol means “is congruent to.”

Reading Math

Course 2

7-2 Angles

An angle is formed by two rayswith a common endpoint. The two rays are the sides of the angle. The common endpoint isis the vertex.

You can name an angle in three ways:

• with the capital letter at the vertex: B,

• with the number inside the angle: 1,

• with three capital letters so that the letter at the vertex is in the middle: ABC or CBA

A

CB

1

Vertex

Course 2

7-2 Angles

Angles are measured in degrees (°). You can use a protractor to measure an angle.

The measure of XYZ is 122°, or m XYZ = 122°.

Course 2

7-2 Angles

An angle’s measure determines the type of angle it is.

A right angle is an angle that that measures exactly 90°. Thesymbol indicates a right angleAn acute angle is an anglethat measures less than 90°

An obtuse angle is an anglethat measures more than 90°but less than180°

A straight angle is an anglethat measures 180°

Course 2

7-2 Angles

If the sum of the measures of two angles is90°, then the angles are complementary angles. If the sum of the measures of twoangles is 180°, then the angles are supplementary angles.

Use the figure to name the following.

Additional Example 3A: Identifying Complementary and Supplementary Angles

Course 2

7-2 Angles

A. one pair of complementary angles

Since 30° + 60° = 90°,

m OMP = 30° and m OMR = 60°

OMP and OMR are complementary.

60° 90°

30°60°

N Q

P

OR

M

Use the figure to name the following.

Try This: Example 3B

Course 2

7-2 Angles

B. one pair of supplementary angles m QMP = 90° and m PMR = 30° + 60° = 90°

Since 90° + 90° = 180°, QMP and PMR are supplementary.

60° 90°

30°60°

N Q

P

OR

M

Course 2

7-3 Parallel and Perpendicular Lines

When lines, segments, or rays intersect, they form angles. If the angles formed by two intersecting lines are equal to 90°, the lines are perpendicular lines.

Some lines in the same plane do not intersect at all. These lines are parallel lines. Segments and rays that are part of parallel lines are also parallel.

Skew lines do not intersect, and yet they are also not parallel. They lie in different planes.

Course 2

7-3 Parallel and Perpendicular Lines

The symbol means “is parallel to.” The symbol means “is perpendicular to.”

Reading Math

Course 2

7-3 Parallel and Perpendicular Lines

Vertical angles are the opposite angles formed by two intersecting lines. When two lines intersect, two pairs of vertical angles are formed. Vertical angles have the same measure, so they are congruent.

Course 2

7-3 Parallel and Perpendicular Lines

A transversal is a line that intersects two or more lines. Eight angles are formed when a transversal intersects two lines. When those two lines are parallel, all of the acute angles formed are congruent, and all of the obtuse angles formed are congruent. These obtuse and acute angles are supplementary.

1 2

3 45 6

7 8

Course 2

7-3 Parallel and Perpendicular Lines

Angles with the same number of tick marks are congruent. The tick marks are placed in the arcs drawn inside the angles.

Reading Math

Line n line p. Find the measure of the angle.

Additional Example 2A: Using Angle Relationships to Find Angle Measures

Course 2

7-3 Parallel and Perpendicular Lines

A. 2 2 and the 130° angle are vertical angles. Since vertical angles are congruent, m 2 = 130°.

Line n line p. Find the measure of the angle.

Additional Example 2B: Using Angle Relationships to Find Angle Measures

Course 2

7-3 Parallel and Perpendicular Lines

B. 3 3 and the 50° angle are acute angles. Since all of the acute angles in the figure are congruent, m 3 = 50°.

Line n line p. Find the measure of the angle.

Additional Example 2C: Using Angle Relationships to Find Angle Measures

Course 2

7-3 Parallel and Perpendicular Lines

C. 4 4 is an obtuse angle. Since all of the obtuse angles in the figure are congruent, m 4 = 130°.