Points, Lines, Planes,
and Angles
5.1
Pre-Algebra
Warm Up
Solve.
1. x + 30 = 90
x = 60
2. 103 + x = 180 x = 77
3. 32 + x = 180
x = 148
4. 90 = 61 + x
x = 29
5. x + 20 = 90
x = 70
Learn to classify and name figures.
Vocabulary
point
line
plane
segment
ray
angle
rightiangle acuteiiangle
obtuseiiangle
complementaryiiangles
supplementaryiiangles
vertical angles
congruent
Points, lines, and planes are the
building blocks of geometry. Segments,
rays, and angles are defined in terms of
these basic figures.
A point names a
location.
•A
Point A
A line is perfectly
straight and
extends forever in
both directions.
l
B
C
line l, or BC
A plane is a
perfectly flat
surface that
extends forever in
all directions.
P
D
E
F
plane P, or
plane DEF
A segment, or
line segment, is
the part of a line
between two
points.
H
G
GH
A ray is a part of
a line that starts
at one point and
extends forever in K
one direction.
J
KJ
Example
A. Name 4 points in the figure.
Point J, point K, Point L, and Point M
B. Name a line in the figure.
KL or JK
Any 2 points on a line can be used.
Example
C. Name a plane in the figure.
Plane
, plane JKL
Any 3 points in the
plane that form a
triangle can be used.
Example
D. Name four segments in the figure.
JK, KL, LM, JM
E. Name four rays in the figure.
KJ, KL, JK, LK
Try This
A. Name 4 points in the figure.
Point A, point B, Point C, and Point D
B. Name a line in the figure.
DA or BC
Any 2 points on a line can be used.
A
D
B
C
Try This
C. Name a plane in the figure.
Plane , plane ABC,
plane BCD, plane CDA,
or plane DAB
Any 3 points in the
plane that form a
triangle can be used.
A
D
B
C
Try This
D. Name four segments in the figure
AB, BC, CD, DA
E. Name four rays in the figure
DA, AD, BC, CB
A
D
B
C
An angle () is formed by two rays with a
common endpoint called the vertex (plural,
vertices). Angles can be measured in degrees.
1
One degree, or 1°, is
of a circle. m1
360
means the measure of 1. The angle can be
named XYZ, ZYX, 1, or Y. The vertex must
be the middle letter.
X
Y
1
Z
m1 = 50°
The measures of angles that fit together to form
a straight line, such as FKG, GKH, and HKJ,
add to 180°.
G
F
H
K
J
The measures of angles that fit together to form
a complete circle, such as MRN, NRP, PRQ,
and QRM, add to 360°.
P
N
M
R
Q
A right angle measures 90°.
An acute angle measures less than 90°.
An obtuse angle measures greater than 90°
and less than 180°.
Complementary angles have measures that
add to 90°.
Supplementary angles have measures that
add to 180°.
Reading Math
A right angle can be labeled with a small box at
the vertex.
Example
A. Name a right angle in the figure.
TQS
B. Name two acute angles in the figure.
TQP, RQS
Example
C. Name two obtuse angles in the figure.
SQP, RQT
Example
D. Name a pair of complementary angles.
TQP, RQS
mTQP + m RQS = 47° + 43° =
90°
Example
E. Name two pairs of supplementary angles.
TQP, RQT m TQP + m RQT = 47° + 133° = 180°
SQP, RQS m SQP + m RQS = 137° + 43° = 180°
Try This
A. Name a right angle in the figure.
BEC
C
B
A
15°
90°
E
75°
D
Try This
B. Name two acute angles in the figure.
AEB, CED
C. Name two obtuse angles in the figure.
BED, AEC
C
B
A
15°
90°
E
75°
D
Try This
D. Name a pair of complementary angles.
AEB, CED mAEB + m CED = 15° + 75° =
90°
C
B
A
15°
90°
E
75°
D
Try This
E. Name two pairs of supplementary angles.
AEB, BED m AEB + m BED = 15° + 165° = 180°
CED, AEC m CED + m AES = 75° + 105° = 180°
C
B
A
15°
90°
E
75°
D
Congruent figures have the same size and shape.
• Segments that have the same length are
congruent.
• Angles that have the same measure are
congruent.
• The symbol for congruence is , which is read “is
congruent to.”
Intersecting lines form two pairs of vertical
angles. Vertical angles are always congruent, as
shown in the next example.
Example
In the figure, 1 and 3 are vertical
angles, and 2 and 4 are vertical angles.
A. If m1 = 37°, find m 3.
The measures of 1 and 2 add to 180° because they
are supplementary, so m2 = 180° – 37° = 143°.
The measures of 2 and 3 add to 180° because they
are supplementary, so m3 = 180° – 143° = 37°.
Example
In the figure, 1 and 3 are vertical
angles, and 2 and 4 are vertical angles.
B. If m4 = y°, find m2.
m3 = 180° – y°
m2 = 180° – (180° – y°)
= 180° – 180° + y°
= y°
Distributive Property
m2 = m 4
Try This
In the figure, 1 and 3 are vertical
angles, and 2 and 4 are vertical angles.
A. If m1 = 42°, find m3.
1
2
4
3
The measures of 1 and 2 add to 180° because they
are supplementary, so m2 = 180° – 42° = 138°.
The measures of 2 and 3 add to 180° because they
are supplementary, so m3 = 180° – 138° = 42°.
Try This
In the figure, 1 and 3 are vertical
angles, and 2 and 4 are vertical angles.
B. If m4 = x°, find m 2.
1
2
4
3
m3 = 180° – x°
m2 = 180° – (180° – x°)
= 180° –180° + x° Distributive Property
m2 = m4
= x°
Lesson Quiz
In the figure, 1 and 3 are vertical angles,
and 2 and 4 are vertical angles.
1. Name three points in the figure.
Possible answer: A, B, and C
2. Name two lines in the figure.
Possible answer: AD and BE
3. Name a right angle in the figure.
Possible answer: AGF
4. Name a pair of complementary angles.
Possible answer: 1 and 2
5. If m1 47°, then find m 3.
47°