Geometry Chapter 2 Test: Reasoning and Proof

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.

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.