Chapter 3 Notes 2
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- Congruent Angle Postulate
- If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.
- Slope of Parallel Line Postulate
- Parallel Lines have the same slope.
- Perpendicular Lines Postulate
- There slope is -1
- Plane and Transversal Postulate
- If two lines in a plane are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.
- Parallel Postulate
- If there is a line and a point not on the lines, then there exists exactly one line through the point that is parallel to the given line.
- Lines with positive slopes, do what?
- Rise
- Lines with negative slopes, do what?
- Fall
- Lines with a slope of zero?
- Horizontal Lines
- Lines with an undefined slope (Example 6/0)?
- Vertical Lines
- Alternative Interior Angles Theorem
- If two parallel lines are cut by a transversal, then each pair of alternative angles is CONGRUENT.
- CONSECUTIVE INTERIOR Angles Theorem
- If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is SUPPLEMENTARY.
- ALTERNATING EXTERIOR Angles Theorem
- If two parallel lines are cut by a tranversal, then each pair of alternate exterior angles is CONGRUENT,
- Perpendicular Transversal Theorem
- In a plane, if a line is PERPENDICULAR to one of two parallel lines, then it is perpendicular to the other.
- Formula for SLOPE
- m = y2-y1/x2-x1
- Skew Lines
- Two lines that do NOT intersect and are NOT in the same plane.
- Parallel Lines
- Two lines in a place that never meet.
- Transversal Line
- a line that intersects two or more lines in a plane at diffferent points.
- Parallel Lines Theorem with Exterior Congruent Angles
- If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel.
- Parallel Lines Theorem with Consecutive Interior Angles (that are supplementary)
- If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is SUPPLEMENTARY, then the lines are parallel.
- Parallel Lines Theorem with Alternate Interior Angles
- If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.
- Parallel Line Theorem with Perpendicular Lines
- In a plane if two lines are perpendicular to the same line, then they are parallel.
- Definition between a Point and a Line
- The distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point.
- Definition of the distance between parallel lines
- Definition of the distance between two parallel lines is the distance between one of the lines and any point on the other line.
- Postulate
- principles accepted as true without proof.
- Conjecture
- Educated Guess
- Counterexample
- An example used to show that a given general state is not always true.
- Conditional Statements
- These are "if" "then" statements.
- Converse
-
Exchanging the hypothesis and conclusion of a conditional .
p q is q p - Negation
- denial of a statement. P represent "not p" or negation of "p"
- Inverse of Conditional Statement
- given a conditional statement its inverse can be formed by negating both the hypothesis and conclusion. The inverse of a true statement is not necessarily true. The inverse of p q is "Not p not q"
- Inductive Reasoning
- when you see the same thing happeneing again and again (a pattern).
- Deductive Reasoning
- When you see the laws of logic and statement that are known to be true reach a conclsuion.
- Law of Detachment
- if p q is a true conditional and p is true than q is true.
- Law of Syllogism
- id p q and q r are true conditionals, then p r is also true.
- Postulates
- Principles accepted as true without proof.
- One Line Postulate
- Through any two points, there is exactly one line.
- One Plane Postulate
- Through any three points NOT on the sme line there is exactly one plane.
- Line Postulate
- A line contains at least two points.
- Place Postulate
- A plane contain at least three points not on the same line. For example Triangle.
- Line and Plane Postulate
- If two point line in a plane, then the entire line containing those two points line in that plane.
- Intersection of Two Planes Postulate
- If two plane intersect, then their INTERSECTION is a line.
- Congruence of Segments Postulate
- Congruence of segments is reflexive, symmetric and transitive.
- Reflexive Property
- a = a
- Symmetric Property
- if a = b then b = a
- Transitive Property
- if a = b and b = c, then a = c
- Addition and Subtraction Property
-
a = b then a + c = b + c and
a - c = b - c - Multiplication and Subtraction Property
-
if a = b then a x c = b x c and
a/c = b/c - Substitution Property
- if a = b then a may be replaced by b in any equation or expression.
- Reflexive Angles
- measure of angle 1 = measure of angle 1
- Symmetric Angles
- if measure of angle A = measure of anlge B then the measure of angle B = the measure of angle A
- Transitive Angles
- if the measure of angle 1 = the measure of agle 2 and the measure of angle 2 = the measure of angle 3, then the measure of angles 1 = measure of angles 3.
- Ruler Postulate
- Points on a line can be matched one to one with set of real numbers.
- Protractor postulate
- Rays from can be matched one for one with real numbers from 0 to 180 degrees.
- Angle Addition Postulate
- If B is in the interior of angle AOC then the measure of angle AOB + BOC = AOC
- Linear Pair Postulate
- If two anlges form a linear pair, then they are supplementary and their sum measures 180 degrees.
- Midpoint Theorem
- On a number line the corrinates of the midpoint of the segment with END points A and B is A + B /2.
- Angles
- Defined by two rays; extend indefinitely in two directions; share a common end point; seperate a plane into three parts(angle, interior, exterior); measured in degrees.
- Math Alert
- a linear pair ALWAYS forms supplementary angles BUT supplemeentary angles do NOT ALWAYS form a linear pair.