Notes

Geometry Notes
Username: A. Pickrell Password: U4me!18p ProctorU: w.-9fo54pp

Lesson 1 – Introduction to Reasoning

Conjecture: An Unproven Statement based on observations

Counterexample: A specific case for which the conjecture is false
Only one counterexample is needed for proving a conjecture wrong

Inductive Reasoning: Making a conjecture for a general case based on specific circumstances

Deductive Reasoning: Uses Facts, knowledge and reasoning to prove a conjecture is true

If-then statements: If a, then b
Converse: If b, then a

Inverse: If not a, then not b

Contrapositive: If not b, then not a

Biconditional: a if and only if b, on can be used when original and converse are both true

Law of Syllogism (A.K.A Chain Rule):
If a then b = true, and b then c = true, then a then c is true

Law of Detachment (Direct Argument):
If the hypothesis (if A, then b) is true, then the conclusion (if a, then B) is true



















Lesson 2 – Points, Lines, and Planes in Space

Point: Location in space with no size
Line: Set of points forming an infinite straight path ---- Undefined and used as material for geometry
Plane: 2D flat surface without an end /

Line Segment: Part of a line that is named by its endpoints

Ray: Part of a line extending infinitely in one direction and centered by an endpoint

If two rays have the same endpoint, they are known as opposite rays
If two points share the same line, they are colinear, and if they share the same plane, they are coplanar


Postulate: An unprovable mathematical statement accepted to be true

Theorem: A mathematical statement proven to be true

Point Postulate: There is only one line in two points

Plane Postulate: There is only one plane in three non-colinear points

Line Postulate: If two points are in a plane, the line containing them is as well

Ruler Postulate: Points on a line can be numbered so every point corresponds to one real number

Named because you can use a ruler to find point distance and is used along with absolute value to do this. Also, when describing distance, line segments do not have a bar accented over the letters like normal notation
Line Intersection Postulate: If two lines intersect, they only do so in one point

Parallel lines are coplanar but do not intersect

Skewed lines are not coplanar and don’t intersect

Plane Intersection Postulate: If two planes intersect, they form a line

Parallel planes, just like parallel lines, do not intersect and are the same distance apart










Lesson 3 – Angles


Angles are formed by two rays, known as sides, with a single endpoint, called the vertex

Can be named by its vertex (if it is clear which angle is named) or its vertex with the addition of the two side endpoints (the endpoints have no order, but the vertex must always be in the middle)

To measure an angle, place a protractor on the angle vertex and measure the two sides at the points on the protractor at which they reside. One side must be at the 0-degree mark and the other at another point on the protractor.

Protractor Postulate:
Given line AB and a point O on line AB. Consider rays OA and OB, as well as all the other rays that can be drawn, with O as an endpoint, on one side of line AB. These rays can be paired with the real numbers between 0 and 180 in such a way that:
1. Ray OA is paired with 0, and ray OB is paired with 180.
2. If ray OR is paired with a and ray OQ is paired with b,
2. then angle ROQ = | a - b |.

Types of Angles
Angle Name Degree Length
Acute angle 0 to 90
Obtuse angle 90 to 180
Reflex angle 180 to 360
Right angle Exactly 90
Straight angle Exactly 180
Zero angle Exactly 0


To Create an angle:
Place a Protractor over the endpoint of a ray and align it with the 0-degree mark. Draw a point at the desired angle and draw a straight edge through the established endpoint of the first ray and the new point to form a new ray.

To create a reflex angle:
Subtract the difference of the reflex angle from 360 and use the angle steps above to create an angle, but instead of marking the inside of the two rays, you mark the outer angle to represent the reflex angle.







Lesson 4 – Midpoint and Properties of Equality
Lesson 5 – Bisectors, Angle Pairs, and Perpendicular Lines
Lesson 6 – Straightedge and Compass Constructions
Lesson 7 – Types of Proofs
Lesson 8 – Basic Proofs
Lesson 9 – Indirect Proofs
Lesson 10 – The Basics of Inequalities
Lesson 11 – Coordinate Geometry
Lesson 12 – Parallel and Perpendicular Lines
Lesson 13 – Parallel Line Proofs
Lesson 14 – Properties of Triangles
Lesson 15 – Triangle Theorems and Proofs
Lesson 16 – Pythagorean Theorem and Special Right Triangles
Lesson 17 – Trigonometric Ratios