Download Notes 38

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Geometry Overview
Vocabulary
Point- 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.
Vocabulary
Angle- 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 line is a straight
path that extends
without end in
opposite directions.
point A
Use a capital
letter to name
a point.
•A
l
X
Y
XY, or YX, or l
Use two points
on the line or a
lowercase letter to
name a line.
Helpful Hint
A number line is an example of a line.
A plane is a
Flat surface that
Has no thickness
and extends
forever.
Q
S
R
plane QRS
Use three points
in any order, not
on the same line,
to name a plane.
Helpful Hint
A coordinate plane is an example of a plane.
Additional Example 1: Identifying Points, Lines, and
Planes
Identify the figures in the diagram.
E
A. three points
D
F
D, E, and F
B. two lines
DE, DF
C. a plane
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
Identify the figures in the diagram.
G
I
A. four points
B. two lines
C. a plane
H
F
A ray is a part of a line.
It has one endpoint and
extends forever
one direction.
A line segment is
part of a line or a ray
that extends from one
endpoint to another.
G
L
H
M
GH
Name the endpoint
first when naming
a ray.
LM, or ML
Use the endpoints
to name a line
segment.
Additional Example 2: Identifying Line Segments and
Rays
Identify the figures in the diagram.
M
N
O
A. three rays
MN, NM, MO
Name the endpoint of
a ray first.
B. two line segments
Use the endpoints in
any order to name a
segment.
MN, MO
Check It Out: Example 2
Identify the figures in the diagram.
A. three rays
D
C
B
B. three line segments
A
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.
Additional Example 3: Identifying Congruent Line
Segments
Identify the line segments that are congruent
in the figure.
AB  CD
One tick mark
AC  BD
Two tick marks
BF  DF  EC  AE Three tick
marks
Reading Math
The symbol  means “is congruent to.”
Check It Out: Example 3
Identify the line segments that are congruent
in the figure.
A
B
C
D
E
A
Vertex
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.
B
1
Angles are measured in degrees (°).
C
An angle’s measure determines the type of
angle it is.
A right angle is an angle that
that measures exactly 90°. The
symbol indicates a right angle.
An acute angle is an angle
that measures less than 90°.
An obtuse angle is an angle
that measures more than 90°
but less than 180°.
A straight angle is an angle
that measures exactly 180°.
Additional Example 1: Classifying Angles
Tell whether each angle is acute, right, obtuse
or straight.
A.
obtuse angle
B.
acute angle
Reading Math
A•
B•
1
•
C
You can name this angle ABC,
CBA, B, or 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 is
90°, then the angles are complementary
angles. If the sum of the measures of two
angles is 180°, then the angles are
supplementary angles.
Additional Example 2A: Identifying Complementary
and Supplementary Angles
Use the diagram to tell whether the angles are
complementary, supplementary, or neither.
OMP and PMQ
To find mPMQ start with the measure that QM
crosses, 105°, and subtract the measure that MP
crosses, 75°. mPMQ = 105° - 75° = 30°. mOMP =
P
60°.
Q
Since 60° + 30° = 90°,
PMQ and OMP are
complementary.
O
N
M
R
Additional Example 2B: Identifying Complementary
and Supplementary Angles
Use the diagram to tell whether the angles are
complementary, supplementary, or neither.
NMO and OMR
mNMO = 15° and mOMR = 165°
P
Since 15° + 165° = 180°,
NMO and OMR are
supplementary.
Reading Math
Read mNMO as
“the measure of
angle NMO.”
Q
O
N
M
R
Additional Example 2C: Identifying Complementary
and Supplementary Angles
Use the diagram to tell whether the angles are
complementary, supplementary, or neither.
PMQ and QMR
To find mPMQ start with the measure that QM
crosses, 105°, and subtract the measure that MP
crosses, 75°. mPMQ = 105° - 75° = 30°. mQMR =
75°.
P
Q
Since 30° + 75° = 105°,
PMQ and QMR are
neither complementary
nor supplementary.
O
N
M
R
Check It Out: Example 2A
Use the diagram to tell whether the angles are
complementary, supplementary, or neither.
BAC and CAF
D
E
C
F
B
A
Additional Example 3: Finding Angle Measures
Angles A and B are complementary. If mA is
56°, what is the mB?
Since A and B are complementary, mA + mB =
90°.
mA + mB = 90°
56° + mB = 90°
– 56°
– 56°
mB = 34°
Substitute 56° for mA.
Subtract 56° from both
sides.
The measure of B = 34°.
Check It Out: Example 3
Angles P and Q are supplementary. If mP is
32°, what is the mQ?