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Transcript
5-1 Points, Lines, Planes, and Angles
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
WARM-UP
Mrs. Meyer’s class is having a pizza party.
Half the class wants pepperoni on the
pizza, 1 of the class wants sausage on the
3
pizza, and the rest want only cheese on
the pizza. What fraction of Mrs. Meyer’s
class wants just cheese on the pizza?
1
6
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.
Plane Geometry Vs Solid Geometry
Can you explane the difference?
explain
YOU NEED TO KNOW THESE
WORDS…see 5-1 in book
point
line
plane
segment
ray
angle
rightiangle
acuteiiangle
obtuseiiangle
complementaryiiangles
supplementaryiiangles
vertical angles
congruent
If you didn’t have time
to write them all down,
don’t bang you head!
Just look in your book
and follow along with
me! You should be
doing this
anyhow!...lesson 5-1
Points, lines, and planes are
the building blocks of
geometry. Segments, rays,
and angles are defined in
terms of these basic
figures. Let’s look more
closely at these things!
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
Additional Example 1A & 1B: Naming Points, Lines,
Planes, Segments, and Rays
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.
Additional Example 1C: Naming Points, Lines, Planes,
Segments, and Rays
C. Name a plane in the figure.
Plane
, plane JKL
Any 3 points in the
plane that form a
triangle can be used.
Additional Example 1D & 1E: Naming Points, Lines,
Planes, Segments, and Rays
D. Name four segments in the figure.
JK, KL, LM, JM
E. Name four rays in the figure.
KJ, KL, JK, LK
Try This: Example 1A & 1B
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: Example 1C
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: Example 1D & 1E
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.
Additional Example 2A & 2B: Classifying Angles
A. Name a right angle in the figure.
TQS
B. Name two acute angles in the figure.
TQP, RQS
Additional Example 2C: Classifying Angles
C. Name two obtuse angles in the figure.
SQP, RQT
Additional Example 2D: Classifying Angles
D. Name a pair of complementary angles.
TQP, RQS
mTQP + m RQS = 47° + 43° =
90°
Additional Example 2E: Classifying Angles
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: Example 2A
A. Name a right angle in the figure.
BEC
C
B
A
15°
90°
E
75°
D
Try This: Example 2B & 2C
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: Example 2D
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: Example 2D & 2E
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.
Additional Example 3A: Finding the Measure of
Vertical Angles
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°.
Additional Example 3B: Finding the Measure of
Vertical Angles
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: Example 3A
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: Example 3B
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°