Geometry: Logic, Proofs, and Conditional Statements Review

Conjecture

A statement that is believed to be true based on observations.

Counterexample

An example that disproves a conjecture.

Inductive Reasoning

Reasoning that involves looking for patterns and making generalizations.

Pattern

A repeated or recurring sequence.

Prediction

A statement about what will happen in the future based on patterns.

Conditional Statement

A statement in the form 'If P, then Q'.

Converse

In symbolic form, if the original statement is "If p, then q", the converse is "If q, then p."

The statement formed by reversing the hypothesis and conclusion of a conditional statement.

Inverse

The statement formed by negating both the hypothesis and conclusion of a conditional statement.

Algebraic Proofs

Proofs that use algebraic properties to show the truth of a statement.

Hypothesis

The part of a conditional statement that follows 'if'.

Conclusion

The part of a conditional statement that follows 'then'.

Biconditional Statement

It asserts that both 'P implies Q' and 'Q implies P'.

A statement that is true in both directions, written as 'P if and only if Q'.

Contrapositive

The statement formed by negating and reversing the converse of a conditional statement. It is logically equivalent to the original statement and can be used in proofs to demonstrate validity.

If not q, then not p.

Law of Detachment

If a conditional statement is true and its hypothesis is true, then its conclusion is also true.

Law of Syllogism

If 'P implies Q' and 'Q implies R' are true statements, then 'P implies R' is also true.

Complementary Angles

Two angles whose measures add up to 90 degrees.

Supplementary Angles

Two angles whose measures add up to 180 degrees.

Adjacent Angles

Two angles that share a common side and vertex.

Vertical Angles

Angles that are opposite each other when two lines intersect.

Linear Pair

A pair of adjacent angles that are supplementary.

This means that their measures add up to 180 degrees.

Assume

To accept something as true without proof for the sake of argument.

Assumption

A statement accepted as true without proof.

Reason

The justification for a conclusion in a proof.

Reasoning

The process of thinking about something in a logical way.

Reasonable

Based on good sense or sound judgment.

Reflexive Property

For any real number a, a = a.

Any real number “a” is always equal to “a”. It is a fundamental property of equality that holds true for all real numbers.

Symmetric Property

If a = b, then b = a.

Commutative Property

The order in which two numbers are added or multiplied does not change the sum or product.

Transitive Property

If a = b and b = c, then a = c.

Addition Property of Equality

If a = b, then a + c = b + c for any real number c.

Multiplication Property of Equality

If a = b, then a c = b c for any real number c.

Substitution Property

If a = b, then a can be replaced by b in any expression.

Distributive Property

a(b + c) = ab + ac.

Truth Value

The classification of a statement as true or false.

Distributive Property of Multiplication Over Addition

The property that states a(b + c) = ab + ac.

Equilateral Triangle

A triangle with three equal sides and three equal angles of 60°.

Parallelogram

A quadrilateral with opposite sides that are equal and parallel.

Square

A parallelogram with four congruent sides and angles.

Converse

Inverse

Contrapostive

Exchange the Hypothesis and the Conclusion: Q — P

Negation of the Hypothesis and the Conclusion: ~P — ~Q
Negation and Exchange of the Hypothesis and the Conclusion: ~Q — ~P

Field Hockey

A team sport played on grass or artificial turf, where players use sticks to hit a ball into the opposing team's goal.

Congruent Angles

Angles that have the same measure.

Fever

An elevated body temperature often indicating illness.

Hospital

A healthcare institution providing treatment and care for patients.

Black Belt

A high rank in martial arts indicating advanced skill.

Brown Belt

A rank in martial arts indicating intermediate skill, typically below black belt.

White Belt

The beginner rank in martial arts.

Conditional Statement Example

If a figure is a square, then it is a rectangle.

Percentage Growth Example

The number of people in the United States who are at least 100 years old grew about 66% from 1980 to 2010.