Translations and Vectors
Vector quantity that can be characterized by its direction and magnitude
Arrow
Length (magnitude) of arrow shows how far to translate
Direction of arrow shows where to translate
B
uuu
r
Vector AB
A = initial point
B = terminal
Composing Reflections Over Parallel Lines
Suppose you reflect a figure over a line, then do it again over a parallel line. You have
done a composition of reflections.
Definition:
The composite of a first transformation S and a second transformation T, den
Reflections Over Intersecting Lines
Rotations composites of 2 reflections over intersecting lines
*magnitude measured in degrees
*direction: clockwise or counterclockwise
Two Reflections Theorem for Rotations
If mintersects l, then the rotation rm o rl :
4-1: Reflecting Points
Geometry (CP)
original point = preimage
To get the image, fold along a line and trace the preimage - gives you the image.
That line is called the reflection line or the line of reflection.
The reflecting line is the perpendicular bi
4-2: Reflecting Figures
Geometry (CP)
There are some properties of figures that are the same in the preimage and the image.
Those properties are preserved by reflections.
Reflection Postulate
Under a reflection:
a) There is a 1-1 correspondence between po
Properties of Angles
5 General Types of Angles
1. Right angle
2. Obtuse angle
3. Acute angle
4. Straight angle
5. Zero angle
If m is the measure of an angle then the angle is:
1. a zero angle if and only if m = 0
2. an acute angle if and only if 0 < m < 9
Perpendicular Lines
Perpendicular lines form a 90 angle at their intersection
1
2
34
Given: m 1 = 90
Prove: m 2 = 90
Statements
0. m 1 = 90
1. m 2 = 90
Reasons
0. Given
1. Linear Pair Theorem
Given: m 1 = 90
Prove: m 4 = 90
Statements
1 = 90
0. m
Parallel Lines
Transversal line that cuts through other lines
Corresponding Angles any pair of angles in a similar location with respect to a
t
transversal and each line
1
3
5
7
2
4
6
l
m
8
Corresponding Angles any pair of angles in a similar location wit
Isometries Notes
Isometry a reflection or a composition of reflections
Two reflection isometries:
1. a reflection over one line = REFLECTION
2. a reflection over 2 parallel lines = TRANSLATION
3. a reflection over intersecting lines = ROTATION
Three refle
One Step Proof Arguments
Proof Argument states the given (p), states the conclusion (q), and shows reasoning of
how to get from p to q.
Algebraic Proof:
If 2x + 3 = 27, then x = 12.
Given: 2x + 3 = 27
Prove: x = 12
Statements
Reasons
0. 2x + 3 = 27
0. Giv
Algebra Properties Used in Geometry
Postulates of Equality
Algebra
Property
Geometry
a=a
1. Reflexive Property of Equality
m A = m A
If a = b, then b = a
2. Symmetric Property of Equality
If m A = m B, then m B = m A
If a = b and b = c,
then a = c
3. Tran