Topic 1: Inductive Reasoning
Definitions:
Inductive Reasoning - reasoning based on patterns you observe
Conjecture - a conclusion you reach using inductive reasoning
Counterexample - an example that shows that a conjecture is incorrect
Helpful Hints:
You only need ONE example to disprove a conjecture, but need to show every case to prove it.
Video Examples:
Inductive Reasoning - reasoning based on patterns you observe
Conjecture - a conclusion you reach using inductive reasoning
Counterexample - an example that shows that a conjecture is incorrect
Helpful Hints:
You only need ONE example to disprove a conjecture, but need to show every case to prove it.
Video Examples:
Topic 2: Conditional Statements
Definitions:
Conditional - is an if-then statement
Hypothesis - is the part (p) following the "if"
Conclusion - is the part (q) following the "then"
p-->q - Read as "if p then q" OR "p implies q"
Biconditional - a single true statement that combines a true conditional and its converse. "if and only if"
Converse: exchanges the hypothesis and conclusion of a conditional (if q, then p)
Inverse: Negates both the hypothesis and conclusion of the conditional (if not p, then not q)
Contrapositive: Negate both the hypothesis and conclusion of the CONVERSE. (If not q, then not p)
Helpful Hints:
Video Examples:
Conditional - is an if-then statement
Hypothesis - is the part (p) following the "if"
Conclusion - is the part (q) following the "then"
p-->q - Read as "if p then q" OR "p implies q"
Biconditional - a single true statement that combines a true conditional and its converse. "if and only if"
Converse: exchanges the hypothesis and conclusion of a conditional (if q, then p)
Inverse: Negates both the hypothesis and conclusion of the conditional (if not p, then not q)
Contrapositive: Negate both the hypothesis and conclusion of the CONVERSE. (If not q, then not p)
Helpful Hints:
Video Examples:
Topic 3: Deductive Reasoning Proofs (algebraic)
Definitions:
Deductive Reasoning - a logical PROCESS in which a conclusion is based on facts that generally assumed to be true. (top-down logic)
Law of Detachment - if the hypothesis of a true conditional is true, then the conclusion is true.
Reflexive Property - a = a
Symmetric Property - if a=b, then b=a
Transitive Property - if a=b and b=c, then a=c
Helpful Hints:
Law of detachment always starts with a very general statment. The next sentence is then a very specific case of that situation.
Law of syllogism is very similar to the transitive property. You can only use it if a hypothesis matches a conclusion, not if both have the same hypothesis or both have the same conclusion.
Video Examples:
Deductive Reasoning - a logical PROCESS in which a conclusion is based on facts that generally assumed to be true. (top-down logic)
Law of Detachment - if the hypothesis of a true conditional is true, then the conclusion is true.
Reflexive Property - a = a
Symmetric Property - if a=b, then b=a
Transitive Property - if a=b and b=c, then a=c
Helpful Hints:
Law of detachment always starts with a very general statment. The next sentence is then a very specific case of that situation.
Law of syllogism is very similar to the transitive property. You can only use it if a hypothesis matches a conclusion, not if both have the same hypothesis or both have the same conclusion.
Video Examples:
Topic 4: Lines
Definitions:
Parallel Lines: two coplanar lines that never intersect. The symbol || means parallel.
Skew Lines: noncoplanar lines, they are not parallel and never intersect
Parallel Planes: planes that do not intersect
Transversal: a line that intersects two or more coplanar lines at distinct points. 8 angles are formed
Helpful Hints:
To visualize skew lines, imagine a line going north to south on the ceiling of the room and another line going east to west on the floor.
Transversals can exist with two or more parallel lines, but also for two or more nonparallel lines
Video Examples:
Parallel Planes and Lines
Parallel Lines: two coplanar lines that never intersect. The symbol || means parallel.
Skew Lines: noncoplanar lines, they are not parallel and never intersect
Parallel Planes: planes that do not intersect
Transversal: a line that intersects two or more coplanar lines at distinct points. 8 angles are formed
Helpful Hints:
To visualize skew lines, imagine a line going north to south on the ceiling of the room and another line going east to west on the floor.
Transversals can exist with two or more parallel lines, but also for two or more nonparallel lines
Video Examples:
Parallel Planes and Lines
Parallel and Skew Lines
Transversals
Topic 5: Angle Pairs
Definitions:
Alternate Interior Angles: nonadjacent interior angles that lie on opposite sides of the transversal
Same-side (consecutive) Interior Angles: interior angles that lie on the same side of the transversal
Corresponding Angles: angles that lie on the same side of the transversal and in corresponding positions.
Alterbate Exterior Angles: nonadjacent exterior angles that lie on opposite sides of the transversal
Helpful Hints:
The angle pair relationships exist whether the two lines cut by a transversal are parallel or not. The relationships of being congruent or supplementary exist ONLY if the lines are parallel.
Corresponding, alternate interior, and alternate exterior angles are all congruent. Only same side interior angles are supplementary.
Video Examples:
Angle Relationships
Alternate Interior Angles: nonadjacent interior angles that lie on opposite sides of the transversal
Same-side (consecutive) Interior Angles: interior angles that lie on the same side of the transversal
Corresponding Angles: angles that lie on the same side of the transversal and in corresponding positions.
Alterbate Exterior Angles: nonadjacent exterior angles that lie on opposite sides of the transversal
Helpful Hints:
The angle pair relationships exist whether the two lines cut by a transversal are parallel or not. The relationships of being congruent or supplementary exist ONLY if the lines are parallel.
Corresponding, alternate interior, and alternate exterior angles are all congruent. Only same side interior angles are supplementary.
Video Examples:
Angle Relationships
More Angle Relationships
Examples Using Angle Relationships
Topic 6: Prove Lines Parallel
Definitions/Theorems:
No new words, so let's focus on theorems
Corresponding Angles Converse: if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
Alternate Interior Angles Converse: If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel
Same-Side Interior Angles Converse: If two lines are cut by a transversal so the same-side interior angles are supplementary, then the lines are parallel
Alternate Exterior Angles Comverse: if two lines are cut by a transversale so the alternate exterior angles are congruent, then the lines are parallel
Helpful Hints:
Solving many of these problems look very similar to the previous lesson. The key thing to remember is here you are showing two lines might be parallel by using the converses.
Video Examples:
No new words, so let's focus on theorems
Corresponding Angles Converse: if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
Alternate Interior Angles Converse: If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel
Same-Side Interior Angles Converse: If two lines are cut by a transversal so the same-side interior angles are supplementary, then the lines are parallel
Alternate Exterior Angles Comverse: if two lines are cut by a transversale so the alternate exterior angles are congruent, then the lines are parallel
Helpful Hints:
Solving many of these problems look very similar to the previous lesson. The key thing to remember is here you are showing two lines might be parallel by using the converses.
Video Examples: