Download Chapter 1 Notes 1, 4, 16, 64, …… -5, -2, 4, 13

Chapter 1 Notes
Section 1.1 – Patterns and Inductive Reasoning
Conjecture- An unproven statement based on observations.
Inductive Reasoning- Looking for patterns and making conjectures.
Inductive Example 1- Draw the next figure
Inductive Example 2
-
Draw the next figure
Describing a number pattern
Inductive Example 3
–
Predict the next number
1, 4, 16, 64, ……
Inductive Example 4
–
Predict the next number
-5, -2, 4, 13, ……
Inductive Example 5
-
Predict the next number
27, 9, 3, 1, …….
Proving Conjectures are …
1. To prove a conjecture is true, you need to ___________________________________
_______________________________________________________________________
2. To prove a conjecture is false, you need to __________________________________
*Counterexample - _____________________________________________________
Example 6
–
Find a counterexample
* The difference between two positive numbers is always positive
Example 7
-
Find a counterexample
* For all real numbers x, the expression x2 is greater than or equal to x.
Goldbach’s Conjecture
two ________________.
– Even numbers greater than two can be written as the sum of
Section 1.2 – Points, Lines, and Planes
Undefined Terms
* _____________ - Has no dimension
* ____________ - One dimension. Extends forever in two directions
* ____________ - Two dimensions. Extends forever. Like a table top.
* Colinear Points – Points that lie on the same line
* Coplanar Points – Points that lie on the same plane
Example 1 –
•
Name 3 points that are collinear
•
Name 4 points that are coplanar
•
Name 3 points that are non-colinear
Important Terms and Symbols
1. Line
2. Line Segment
3. Ray
4. Opposite Rays
Example 2 –
*
*
*
*
Draw 3 non-colinear points and label them A, B, and C
Draw line AB
Draw segment BC
Draw ray CA
Intersections of Lines and Planes
•
Two or more geometric figures intersect if they have one or more points in
common
•
Two ________________ intersect at one _________________
•
Two ________________ intersect at one _________________
Section 1.3 – Segments and Their Measures
Postulate – A rule accepted without ______________________
**Remember that distance is always _________________________
Length/ Distance
- Notation: ____
Segment Addition Postulate – If B is between A and C, then
_____________________________
B
A
Seg. Add. Post. Example 1-
C
Suppose M is between L and N. Use the segment
addition postulate to solve for x. Then find the lengths of LM, MN, and LN.
L
a. LM = 3x + 8
MN = 2x – 5
LN = 23
M
N
b. LM = ½ z + 2
MN = 3z + 1.5
LN = 5z + 2
TRY -
Identify the unknown lengths given that BD = 4, AE = 17,
AD = 7, and BC = CD
A
B
C
D
E
***** Distance Formula ****
If A(x1 , y1) and B(x2 , y2) are points in a
coordinate plane, then the distance between
points A and B is
Distance Formula Example 2 – Find the distance between points
O and K.
O( 2, 6)
K( 5, 10)
Distance Formula Example 3 – Find the distance between points
A and C.
A( -4, 7)
C( 3, -2)
Section 1.4 – Angles and Their Measures
What is an angle?
*Consists of two different ____________ with the same
___________________.
Naming an angle
* All angles are named using three points
- The ___________ must be in the middle
Types of angles
Name
Description
Picture
Acute
Right
Obtuse
Straight
Postulate 4: Angle Addition Postulate
This postulate allows you to add the two smaller angles together to find the measure of
the larger angle.
Example
Adjacent Angles- Two angles are adjacent if they share a common vertex and side, but
have no common interior points.
Example
Section 1.5 – Segment and Angel Bisectors
* Midpoint – The point that divides a segment into two congruent segments
* Segment Bisector – A segment, ray, line, or plane that intersects the original segment
at the midpoint.
Midpoint Formula
Midpoint Example 1 - How to find the midpoint given the two endpoints
A(1,2) B(7,10)
Midpoint Example
-
How to find the missing endpoint given one end point and the
midpoint
(8,5) is midpoint and (4, 9) is one endpoint
Angle Bisector – A ray that divides an angle into two adjacent angles that are congruent
Angle Bisector Example 5a –
Angle Bisector Example 5b
–
Section 1.6 – Angle Pair Relationships
* Vertical Angles -
Two angles are vertical angles if their sides form two pairs of
_______________________________.
3
1
2
4
***VERTICAL ANGLE PAIRS ARE ALWAYS CONGRUENT
Linear Pair – Two adjacent angles whose non-common sides are opposite rays.
The sum the two angles is 180 degrees, this is because they form a straight line.
Example 1 – Solve for x and y
Example 2 – Solve for x and y
Complementary v. Supplementary
Complementary- Two angles whose sum is ____________
Supplementary- Two angles whose sum is _____________
ExampleFind ∠A, ∠B, and ∠C if ∠A is supplementary to ∠B and ∠A is complementary to ∠C
Section 1.7 – Perimeter, Area, and Circumference
Square
Triangle
Rectangle
Circle