Unit 2

Inductive Reasoning

Making a general conclusion based on a pattern of specific examples or observations.

Conjecture

An unproven statement or conclusion reached by using inductive reasoning.

Statement

A sentence that is either strictly true ($T$) or false ($F$), but not both.

Negation

The statement that has the opposite truth value of the original statement. Represented by the tilde symbol: $\sim p$ (read as "not $p$").

Compound Statement

A statement formed by combining two or more individual statements using the words "and" or "or".

Conjunction

A compound statement joined by "and". Represented by $\land$ (e.g., $p \land q$). It is only True if both individual statements are true.

Disjunction

A compound statement joined by "or". Represented by $\lor$ (e.g., $p \lor q$). It is True if at least one of the individual statements is true.

Truth Table

A visual matrix used to determine the truth or falsehood of a compound statement based on every possible scenario of its individual parts.

Conditional Statement

An "if-then" statement. Represented by an arrow: $p \rightarrow q$ (read as "$p$ implies $q$").

Hypothesis

The "if" part of a conditional statement (the $p$ in $p \rightarrow q$). It states the condition.

Conclusion

The "then" part of a conditional statement (the $q$ in $p \rightarrow q$). It states the outcome.

Conditional Statement Truth Table Rule

A conditional statement is always True except when a true hypothesis leads to a false conclusion ($T \rightarrow F$ is False).

Ruler Postulate

The points on any line can be matched one-to-one with real numbers (coordinates), allowing us to measure the exact distance between points.

Segment Addition Postulate

If point $B$ lies between points $A$ and $C$ on a line, then the parts equal the whole: $AB + BC = AC$.

Proof Table (Two-Column Proof)

A formal logical layout containing a numbered column of "Statements" (mathematical facts/claims) and a parallel column of "Reasons" (definitions, postulates, or theorems that justify each claim).

Angle Addition Postulate

If point $D$ lies in the interior of $\angle ABC$, then the two smaller adjacent angles add up to the larger angle: $m\angle ABD + m\angle DBC = m\angle ABC$.

Supplement Theorem

If two angles form a linear pair, then they are automatically supplementary ($m\angle 1 + m\angle 2 = 180^\circ$).

Complement Theorem

If the noncommon sides of two adjacent angles form a right angle, then the angles are automatically complementary ($m\angle 1 + m\angle 2 = 90^\circ$ ).

Reflexive Property of Congruence

Any geometric figure is congruent to itself (e.g., $\overline{AB} \cong \overline{AB}$ or $\angle A \cong \angle A$).

Symmetric Property of Congruence

If one figure is congruent to a second, then the second is congruent to the first (e.g., If $\angle A \cong \angle B$, then $\angle B \cong \angle A$).

Transitive Property of Congruence

If figure 1 is congruent to figure 2, and figure 2 is congruent to figure 3, then figure 1 is congruent to figure 3 (e.g., If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$).

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles), then those two angles are congruent to each other (e.g., If $\angle 1 + \angle 2 = 180^\circ$ and $\angle 3 + \angle 2 = 180^\circ$, then $\angle 1 \cong \angle 3$).

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles), then those two angles are congruent to each other (e.g., If $\angle 1 + \angle 2 = 90^\circ$ and $\angle 3 + \angle 2 = 90^\circ$, then $\angle 1 \cong \angle 3$).