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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
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