Download Algebra 2B Notes

Geometry
Chapter 1 Notes
1.1 Patterns and Inductive Reasoning
Conjecture – An unproven statement
Inductive Reasoning – The process of making conjectures from patterns.
Counterexample – An example that shows a conjecture is false.
Patterns
1, 2, 4, 7, 11, ……….
add 1, add 2, add 3, add 4, etc
3, 0, -3, -6 …….
Subtract3, subtract 3, subtract 3, etc.
1.2
Points, Lines, and Planes
A point has no dimension
●A
A line extends in one dimension and symbolized or written: AB
●
●
A
B
A plane extends in two dimensions
Collinear points are points that lie on the same line
●
●
●
A
B
C
Coplanar points that lie on the same plane
●
●
●
A line segment consists of endpoints on the same line and all the points
between and symbolized or written as:
AB
●
●
A
B
A ray consists of an initial point A and consists of all the points on the line
AB that lie on the same side of A as the point B and is symbolized or
written as:
●
●
A
B
AB
Opposite rays have the same initial point but extend in opposite directions
●
A
Example: CA and CB are opposite rays
●
C
.H
Examples:
.
E
A, B, C are collinear
E, B, C are not collinear
C is not on ray BA
BE and BD are opposite rays
A, B, G are coplanar
AB H are coplanar
.A
.
B
C
.
D
.
1.3
.G .
J
Segments and Their Measures
Postulates or Axioms are rules that are accepted without proof
Segment Addition Postulate
If B is between A and C then AB + BC = AC
●
A
●
B
●
C
Example: Find the lengths of AB & BC
AB = 3x + 8
BC = 2x – 5
3x + 8 + 2x – 5 = 23
5x + 3 = 23
5x = 20
x=4
AC = 23
Therefore AB = 3(4) + 8 = 20
BC = 2(4) – 5 = 3
●
B
Distance Formula
AB =
(x2 – x1)2 + ( y2 – y1)2
Example: A( -2, -6) and B(1, -2)
AB =
1.4
(1 - -2)2 +(-2 - - 6)2 =
32 + 42 =
25
= 5
Angles and Their Measures
An angle consists of two different rays that have the same initial point and is
written as
BAC or
CAB or A
C
●
Sides are the rays of the angle (AC and AB)
The vertex is the initial point of the angle. (A)
●
A
B
Congruent Angles
Angles with the same measure are called congruent angles written as:
 ABC 
 DEF
Measure of an Angle
The measure of an angle is written in degrees ( °) and can be approximated
with a protractor. Example: Angle ABC is 50°
C
B
A
Interior/Exterior points
C
Exterior
●
interior
●
A
B
Angle Types
Acute
= less than 90°
Right
= 90°
Obtuse
= greater than 90°
Straight = 180°
Angle Addition Postulate
D
 ABC +  DBC
=  ABD
C
B
1.5
A
Segment and Angle Bisectors
Midpoint – The midpoint of a segment is the point that divides or bisects the
segment into two congruent segments. (B)
●
A
●
B
●
C
M = (x2 + x1) , ( y2 + y1)
2
2
Segment bisector – A segment bisector is a segment, ray, line, or plane that
intersects a segment at its midpoint. (AB = BC)
●
A
●
B
●
C
Angle bisector – A angle bisector is a ray that divides an angle into two
adjacent angles that are congruent. ABC   CBD
D
C
B
A
Examples: BC is a angle bisector of ABD, find the value of x if
ABC = (3x - 20°) and CBD = (x + 40°).
3x - 20 = x + 40
-x
-x
2x – 20 = 40
+20 = 20
2x = 60
x = 30
1.6
Angle Pair Relationships
Vertical Angles – Two angles are vertical angles if their sides form two pairs
opposite rays
Linear pair – Two adjacent angles are a linear pair if their noncommon sides
are opposite rays
Finding angle measures
Complementary Angles (Sum equals 90°)
ABC and  CBD are complementary
if ABC + CBD = 90°
D
C
Supplementary angles (Sum equals 180°)
ABC and  CBE are supplementary
if ABC + CBE = 180°
Examples:
B
E
A
AEC +  CEB
D
= 180°
E
( y + 20)°
(y + 20) + (4y - 15 ) = 180
(4y – 15)°
B
C
5y + 5 = 180
A
5y = 180
y = 35
1.7
Introduction to Perimeter, Circumference, and Area
l
Perimeter = The sum of all the sides
w
Area
Rectangle
Triangle
Circle
-
A= length x width (A = lw)
A = ½ base x height (A = ½bh)
A = πr2
Circumference
Circle
-
C = 2πr
h
b
π = 3.14
●
r