Geometry Legal Reasons
Geometry Legal Reasons
Level 3
Angles, Arcs, Equality Postulates, Slope, Parallel Lines, Perpendicular Lines
How to Play | Game Levels | Reasons | Cards | Game Board | Writing Proofs
1
Definition of Angle An angle is the union of two rays that have the same endpoint.
Point A is the VERTEX and the rays are the SIDES.
2
The Angle Measure Postulate
  • Unique Measure Assumption: Every angle has a unique measure from zero degrees to 180 degrees.
  • Unique Angle Assumption:
  • Zero Angle Assumption:

  • Straight Angle Assumption:

  • Angle Addition Property:
3
Definition of Angle Bisector
4
Definition of Central Angle of a Circle
5
Definitions of Measure of Minor and Major Arcs

*Note: The degree measure of an arc indicates an amount of turn and is different than arc length which is an actual distance, or portion of the circumference of a circle.

6 Rotations:

Rotations are turns. They are measured with arc measure, and the direction of a rotation is determined by its sign.

Clockwise rotations are always negative (-).
Counterclockwise rotations are always positive (+).

*Note: The magnitude of a rotation can be negative, but the magnitude of an angle (by the Angle Measurement Postulate) can't be negative.

Rotations also can have magnitudes outside the range of -180 to 180 degrees, such as 400 degrees. This can always be converted to a rotation in the given range by adding or subtracting a multiple of 360 degrees. A rotation of 400 degrees will land the user in the same spot as one of 40 degrees.

7

 

Definitions of Types of Angles

If m is the measure of an angle, then the angle is:

  • zero if and only if m=0
  • acute if and only if 0 < m < 90
  • right if and only if m = 90
  • obtuse if and only if 90 < m < 180
  • straight if and only if m = 180
8
Definitions of Complementary and Supplementary Angles

If the measures of two angles are s and t, then the angles are:

  • complementary if and only if s +t = 90
  • supplementary if and only if s + t =180
9
Definition of Adjacent Angles
10
Definition of Linear Pair Two adjacent angles form a linear pair if and only if their non-common sides are opposite rays.
11
Linear Pair Theorem If two angles form a linear pair, then they are supplementary.
12
Definition of Vertical Angles Two non-straight angles are vertical angles if and only if the union of their sides is two lines.
Angles 4 and 2 are vertical, and so are angles 1 and 3.
13
Vertical Angles Theorem If two angles are vertical angles, then they have equal measures.
14
Postulates of Equality

For any real numbers a, b, and c:

  • Reflexive Property of Equality: a = a.
  • Symmetric Property of Equality:
    If a = b, then b = a.
  • Transitive Property of Equality:
    If a= b and b = c, then a = c.
15
Equation to Inequality Property If a and b are positive, real, numbers and a + b = c, then c > a and c > b.
16
Corresponding Angles Postulate

Suppose two coplanar lines are cut by a transversal.

  • If two corresponding angles have the same measure, then the lines are parallel.
  • If the lines are parallel, then corresponding angles have the same measure.
17
Definition of Slope

When a linear equation is written in "slope intercept form" it is solved for y and looks like this: y = mx + b. The number "m", in front of the x, is the slope, or steepness of the line. The number in the "b" spot is the y intercept, the place where the line crosses the y axis.

HORIZONTAL LINES have zero slope.
VERTICAL LINES have undefined slope.
LINES WITH NEGATIVE SLOPE appear to be going "downhill" as you look at them from left to right.
LINES WITH POSITIVE SLOPE appear to be going "uphill" as you look at them from left to right.

18
Parallel Lines and Slopes Theorem Two nonvertical lines are parallel if and only if they have the same slope.
19
Transitivity of Parallelism Theorem In a plane, if line s is parallel to line m and line m is parallel to line t, then line s is parallel to line t.
20
Definition of Perpendicular Two segments, rays, or lines are perpendicular if and only if the lines containing them form a 90 degree angle.
21
Two Perpendiculars Theorem If two coplanar lines a and b are each perpendicular to the same line, then they are parallel to each other.
22
Perpendicular to Parallels Theorem In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
23
Perpendicular Lines and Slopes Theorem Two nonvertical lines are perpendicular if and only if the product of their slopes is -1.
How to Play | Game Levels | Reasons | Cards | Game Board | Writing Proofs
Home | About Us | Algebra| Dictionary | Games | Geometry | Gym | Humor | Lab | Magic | Natural Math | PreAlgebra | Resources | Teachers Only | Toolbox | Treasures | Videos | Wonders | Writings

Copyright © 1999-2020 themathlab.com


Google