Geometry Overview

*Notes 38

VocabularyPoint- an exact location. It is usually represented as a dot, but it has no size at all.Line- a straight path that extends without end in

opposite directions.Plane- a flat surface that has no thickness and extends forever.Ray- a part of a line. It has one endpoint and extends forever one direction.Line segment- part of a line or a ray that extends from one endpoint to another.Congruent- figures that have the same shape and

size. Line segments are congruent if they have the same length.

VocabularyAngle- formed by two rays with a common endpoint.Vertex- the common endpoint of an angle where the

two rays meet.Right angle- an angle that measures exactly 90°. Acute angle- an angle that measures less than 90°.Obtuse angle- an angle that measures more than 90°

but less than 180°.Straight angle- an angle that measures exactly 180°.Complementary angles- sum of the measures of two

angles is 90°Supplementary angles- sum of the measures of two

angles is 180°

A point is an exact location. 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 YX, or lUse two pointson the line or a lowercase letter to name a line.

X Y

l

A plane is a Flat surface that Has no thickness and extends forever.

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

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.

Check It Out: Example 1

HI

A. four points

B. two lines

C. a plane

Identify the figures in the diagram.

G

F

A ray is a part of a line.It has one endpoint andextends foreverone 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

Identify the figures in the diagram.

Additional Example 2: Identifying Line Segments and Rays

M N

O

A. three rays

B. two line segments

MN, NM, MO

MN, MO

Name the endpoint of a ray first.

Use the endpoints in any order to name a segment.

Identify the figures in the diagram.

C

AB

D

Check It Out: Example 2

A. three rays

B. three line segments

Figures are congruent if they have the same shape and size. Line segments are congruent if they have the same length.

You can use tick marks to indicate congruent line segments. In the triangle below, line segments AB and BC are congruent.

Identify the line segments that are congruent in the figure.

Additional Example 3: Identifying Congruent Line Segments

AB CD

AC BD

BF DF EC AE

One tick mark

Two tick marks

Three tick marks

The symbol means “is congruent to.”

Reading Math

Check It Out: Example 3

Identify the line segments that are congruent in the figure.

A

B C

D E

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

Angles are measured in degrees (°).

A

CB

1

Vertex

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 angle.An acute angle is an anglethat measures less than 90°.

An obtuse angle is an anglethat measures more than 90°but less than 180°.

A straight angle is an anglethat measures exactly 180°.

Tell whether each angle is acute, right, obtuse or straight.

Additional Example 1: Classifying Angles

A. B.

obtuse angle acute angle

You can name this angle ABC, CBA, B, or 1.

Reading Math

A •

B • • C

1

Check It Out: Example 1

Tell whether each angle is acute, right, obtuse, or straight.

A. B.

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 diagram to tell whether the angles are complementary, supplementary, or neither.

Additional Example 2A: Identifying Complementary and Supplementary Angles

OMP and PMQ

Since 60° + 30° = 90°, PMQ and OMP are complementary. O

N

P Q

RM

To find mPMQ start with the measure that QM crosses, 105°, and subtract the measure that MP crosses, 75°. mPMQ = 105° - 75° = 30°. mOMP = 60°.

Use the diagram to tell whether the angles are complementary, supplementary, or neither.

Additional Example 2B: Identifying Complementary and Supplementary Angles

NMO and OMR

mNMO = 15° and mOMR = 165°

O

N

P Q

RM

Since 15° + 165° = 180°, NMO and OMR are supplementary.

Read mNMO as “the measure of angle NMO.”

Reading Math

Use the diagram to tell whether the angles are complementary, supplementary, or neither.

Additional Example 2C: Identifying Complementary and Supplementary Angles

PMQ and QMR

O

N

P Q

RM

Since 30° + 75° = 105°, PMQ and QMR are neither complementary nor supplementary.

To find mPMQ start with the measure that QM crosses, 105°, and subtract the measure that MP crosses, 75°. mPMQ = 105° - 75° = 30°. mQMR = 75°.

Use the diagram to tell whether the angles are complementary, supplementary, or neither.

Check It Out: Example 2A

BAC and CAF

C

B

D

E

FA

Angles A and B are complementary. If mA is 56°, what is the mB?

Additional Example 3: Finding Angle Measures

Since A and B are complementary, mA + mB = 90°.

mA + mB = 90°

56° + mB = 90°

– 56° – 56°

mB = 34°

Substitute 56° for mA.

Subtract 56° from both sides.

The measure of B = 34°.

Angles P and Q are supplementary. If mP is 32°, what is the mQ?

Check It Out: Example 3