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Pre-Algebra
5-1 Points, Lines, Planes, and Angles
5-1 Points, Lines, Planes, and Angles
Pre-Algebra
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Warm UpSolve.
1. x + 30 = 90
2. 103 + x = 180
3. 32 + x = 180
4. 90 = 61 + x
5. x + 20 = 90
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
x = 60
x = 77
x = 148
x = 29
x = 70
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Problem of the Day Mrs. Meyer’s class is having a pizza party. Half the class wants pepperoni on the pizza, of the class wants sausage on the 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
13
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Vocabularypoint line plane
segment ray angle
rightiangle acuteiiangle
obtuseiiangle complementaryiiangles
supplementaryiiangles
vertical angles
congruent
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures.
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
A line is perfectly straight and extends forever in both directions.
line l, or BCB
Cl
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
A plane is a perfectly flat surface that extends forever in all directions.
plane P, or plane DEF
DE
F
PP
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
G
HA segment, or line segment, is the part of a line between two points.
GH
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
K
JA ray is a part of a line that starts at one point and extends forever in one direction.
KJ
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Additional Example 1A & 1B: Naming Points, Lines, Planes, Segments, and Rays
A. Name 4 points in the figure.
B. Name a line in the figure.
Point J, point K, Point L, and Point M
Any 2 points on a line can be used.KL or JK
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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.
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Additional Example 1D & 1E: Naming Points, Lines, Planes, Segments, and Rays
D. Name four segments in the figure.
E. Name four rays in the figure.KJ, KL, JK, LK
JK, KL, LM, JM
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Try This: Example 1A & 1B
A. Name 4 points in the figure.
B. Name a line in the figure.
Point A, point B, Point C, and Point D
A B
CD
DA or BC Any 2 points on a line can be used.
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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 B
CD
Try This: Example 1C
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
D. Name four segments in the figure
E. Name four rays in the figureDA, AD, BC, CB
AB, BC, CD, DA
A B
CD
Try This: Example 1D & 1E
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
An angle () is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees.
One degree, or 1°, is of a circle. m1
means the measure of 1. The angle can be named XYZ, ZYX, 1, or Y. The vertex must be the middle letter.
1360
X
Y Z1 m1 = 50°
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
The measures of angles that fit together to form a straight line, such as FKG, GKH, and HKJ, add to 180°.
F K J
G H
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
The measures of angles that fit together to form a complete circle, such as MRN, NRP, PRQ, and QRM, add to 360°.
P
R QM
N
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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°.
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
A right angle can be labeled with a small box at the vertex.
Reading Math
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Additional Example 2A & 2B: Classifying Angles
A. Name a right angle in the figure.
B. Name two acute angles in the figure.
TQS
TQP, RQS
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Additional Example 2C: Classifying Angles
C. Name two obtuse angles in the figure.
SQP, RQT
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Additional Example 2D: Classifying Angles
D. Name a pair of complementary angles.
TQP, RQS mTQP + mRQS = 47° + 43° = 90°
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Additional Example 2E: Classifying Angles
E. Name two pairs of supplementary angles.
TQP, RQT
SQP, RQS
mTQP + mRQT = 47° + 133° = 180°
mSQP + mRQS = 137° + 43° = 180°
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
Try This: Example 2A
A. Name a right angle in the figure.
BEC
E
D
CB
A 90° 75°15°
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
C. Name two obtuse angles in the figure.
BED, AEC
B. Name two acute angles in the figure.
AEB, CED
E
D
CB
A 90° 75°15°
Try This: Example 2B & 2C
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
D. Name a pair of complementary angles.
AEB, CED
E
D
CB
A 90° 75°15°
Try This: Example 2D
mAEB + mCED = 15° + 75° = 90°
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
E. Name two pairs of supplementary angles.
AEB, BED
CED, AEC
E
D
CB
A 90° 75°15°
Try This: Example 2D & 2E
mAEB + mBED = 15° + 165° = 180°
mCED + mAES = 75° + 105° = 180°
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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.
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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 m1 = 37°, find m3.
The measures of 1 and 2 add to 180° because they are supplementary, so m2 = 180° – 37° = 143°.
The measures of 2 and 3 add to 180° because they are supplementary, so m3 = 180° – 143° = 37°.
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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 m4 = y°, find m2.
m3 = 180° – y°
m2 = 180° – (180° – y°)
= 180° – 180° + y°
= y°
Distributive Property m2 = m4
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles.
A. If m1 = 42°, find m3.
The measures of 1 and 2 add to 180° because they are supplementary, so m2 = 180° – 42° = 138°.
The measures of 2 and 3 add to 180° because they are supplementary, so m3 = 180° – 138° = 42°.
12
34
Try This: Example 3A
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
In the figure, 1 and 3 are vertical angles, and 2 and 4 are vertical angles.
B. If m4 = x°, find m2.
m3 = 180° – x°
m2 = 180° – (180° – x°)
= 180° –180° + x°
= x°
Distributive Property m2 = m4
12
34
Try This: Example 3B
Pre-Algebra
5-1 Points, Lines, Planes, and Angles
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.
3. Name a right angle in the figure.
4. Name a pair of complementary angles.
5. If m1 47°, then find m3.
2. Name two lines in the figure.
Possible answer: A, B, and C
Possible answer: AGF
Possible answer: 1 and 2
47°
Possible answer: AD and BE