Geometry Chapter 2 Test: Reasoning and Proof

Questions and Answers

What is a conditional?

A statement that shows something is false

A property of equality

An if-then statement (correct)

A conclusion drawn from a hypothesis

What is the hypothesis in a conditional?

The part of the conditional that comes after the word 'if'

What is the conclusion in a conditional?

The part of the conditional that comes after the word 'then'

What defines a counterexample?

An example that shows that a conjecture is incorrect Signup and view all the answers

What is a converse in logic?

Switch the hypothesis and the conclusion in a conditional Signup and view all the answers

What is the negation of a statement?

The opposite of the statement Signup and view all the answers

What is the inverse of a conditional?

Negate both the hypothesis and the conclusion of the conditional Signup and view all the answers

What does the contrapositive of a conditional do?

Switches the hypothesis and the conclusion and negates both Signup and view all the answers

What is a biconditional?

A single true statement that combines a true conditional and its true converse using 'if and only if' Signup and view all the answers

What constitutes a good definition?

Uses clearly understood terms, is precise and is reversible Signup and view all the answers

What does the Law of Detachment state?

If a conditional statement is true and its hypothesis is true, then its conclusion is true Signup and view all the answers

What is the Law of Syllogism?

Takes two conditional statements and forms a conclusion by combining the hypothesis of one statement and the conclusion of another Signup and view all the answers

What does the Addition Property of Equality state?

If a = b, then a + c = b + c Signup and view all the answers

What does the Subtraction Property of Equality state?

If a = b, then a - c = b - c Signup and view all the answers

What does the Multiplication Property of Equality state?

If a = b, then a * c = b * c Signup and view all the answers

What does the Division Property of Equality state?

If a = b and c ≠ 0, then a/c = b/c Signup and view all the answers

What does the Reflexive Property of Equality state?

a = a Signup and view all the answers

What does the Symmetric Property of Equality state?

If a = b, then b = a Signup and view all the answers

What does the Transitive Property of Equality state?

If a = b and b = c, then a = c Signup and view all the answers

What does the Substitution Property of Equality state?

If a = b, then b can replace a in any expression Signup and view all the answers

What does the Distributive Property of Equality state?

a(b + c) = ab + ac Signup and view all the answers

What does the Reflexive Property of Congruence state?

∠A ≅ ∠A Signup and view all the answers

What does the Symmetric Property of Congruence state?

If ∠A ≅ ∠B, then ∠B ≅ ∠A Signup and view all the answers

What does the Transitive Property of Congruence state?

If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C Signup and view all the answers

What does Thm. 2-1: Vertical Angles Theorem state?

Vertical angles are congruent Signup and view all the answers

What are supplementary angles?

Two angles whose sum is 180 degrees Signup and view all the answers

What are complementary angles?

Two angles whose sum is 90 degrees Signup and view all the answers

What does Thm. 2-4 state?

All right angles are congruent Signup and view all the answers

What does Thm. 2-2: Congruent Supplements Theorem state?

If two angles are supplementary to the same angle or congruent angles, then the two angles are congruent Signup and view all the answers

What does Thm. 2-5 state?

If two angles are congruent and supplementary, then each is a right angle Signup and view all the answers

What does Thm. 2-3: Congruent Complements Theorem state?

If two angles are complementary to the same angle (or of congruent angles), then the two angles are congruent Signup and view all the answers

Study Notes

Conditional Statements and Logic

A conditional is an if-then statement that establishes a cause-and-effect relationship.
The hypothesis follows "if" while the conclusion follows "then."
The converse switches the hypothesis and conclusion.
Negation represents the opposite of a statement.
The inverse negates both the hypothesis and conclusion of a conditional.
The contrapositive switches and negates both components of a conditional.
A biconditional statement combines a conditional and its converse, valid only if both are true.

Definitions and Properties

Good definitions must use clear terms, be precise, and reversible, resulting in a biconditional statement.
The Law of Detachment states that if a conditional is true and its hypothesis is confirmed, the conclusion is also true.
The Law of Syllogism allows for concluding a new statement by linking two true conditional statements.

Properties of Equality

Addition Property: If a = b, then a + c = b + c.
Subtraction Property: If a = b, then a - c = b - c.
Multiplication Property: If a = b, then a * c = b * c.
Division Property: If a = b and c ≠ 0, then a/c = b/c.
Reflexive Property: a = a, establishing that any quantity is equal to itself.
Symmetric Property: If a = b, then b = a, highlighting equality's bidirectional nature.
Transitive Property: If a = b and b = c, then a = c, confirming consistency in equality.
Substitution Property: If a = b, b can replace a in expressions.
Distributive Property: a(b + c) = ab + ac demonstrates distributing multiplication over addition.

Properties of Congruence

Reflexive Property of Congruence states that an angle is congruent to itself (∠A ≅ ∠A).
Symmetric Property states if ∠A ≅ ∠B, then ∠B ≅ ∠A.
Transitive Property states if ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.

Angle Theorems

Vertical angles are congruent, outlined in the Vertical Angles Theorem (Thm. 2-1).
Supplementary angles add up to 180 degrees, while complementary angles sum to 90 degrees.
All right angles are congruent (Thm. 2-4).
Congruent Supplements Theorem (Thm. 2-2) asserts that angles supplementary to the same angle are congruent.
If two angles are congruent and supplementary, they are each a right angle (Thm. 2-5).
Congruent Complements Theorem (Thm. 2-3) states that angles complementary to the same angle are congruent.

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Test your understanding of key concepts in Geometry Chapter 2, focusing on reasoning and proof. This quiz includes important terms such as conditional, hypothesis, conclusion, and counterexample, helping you solidify your foundational knowledge.