Warm Up

1
Warm Up Solve. 1. x 30 90 2. 103 x
180 3. 32 x 180 4. 90 61 x 5. x 20
90
x 60
x 77
x 148
x 29
x 70
2
Points, Lines, Planes, and Angles

• Wednesday, November 11, 2009

3
Points, lines, and planes

• Points, lines, and planes are the building blocks
  of geometry.
• Segments, rays, and angles are defined in terms
  of these basic figures.

4
Definitions
5
Definitions
6
Examples
A. Name 4 points in the figure.
Point J, point K, point L, and point M
B. Name a line in the figure.
Any 2 points on a line can be used.
C. Name a plane in the figure.
Any 3 points in the plane that form a triangle
can be used.
7
Examples
D. Name four segments in the figure
E. Name four rays in the figure
A
B
C
D
8
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. m?1
• The angle can be named ?XYZ, ?ZYX, ?1, or ?Y. The
  vertex must be the middle letter.

9
The measures of angles that fit together to form
a complete circle, such as ?MRN, ?NRP, ?PRQ, and
?QRM, add to 360.
10
Types of 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.

11
The measures of angles that fit together to form
a straight line, such as ?FKG, ?GKH, and ?HKJ,
add to 180.
12
Examples
A. Name a right angle in the figure.
?TQS
B. Name two acute angles in the figure.
?TQP, ?RQS
C. Name two obtuse angles in the figure.
?SQP, ?RQT
D. Name a pair of complementary angles.
m?TQP m? RQS 47 43 90
?TQP, ?RQS
E. Name two pairs of supplementary angles.
?TQP, ?RQT
m?TQP m? RQT 47 133 180
m?SQP m? SQR 137 43 180
?SQP, ?SQR
13
Congruent

• 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.

14
Example
In the figure, ?1 and ?3 are vertical angles, and
?2 and ?4 are vertical angles.
If m?1 37, find m? 3.
Vertical angles are congruent so m?1 m?3.
m?3 37
Find m? 2.
?2 and ?3 are supplementary.
m?2 180 37 143