Geometry Chapter 2 Test: Reasoning and Proof
Thm. 2-1: Vertical Angles Theorem
Vertical angles are congruent. Example: ∠A ≅ ∠B
Reflexive Property of Equality
a = a
Distributive Property of Equality
a(b + c) = ab + ac
supplementary angles
Two angles whose sum is 180 degrees
Good definition
A good definition uses clearly understood terms, is precise and is reversible. Both the conditional and its converse must be true for it to be a good definition. It can be written as a biconditional. Example: Conditional:If points are collinear then they lie on the same line. ~true Converse: If points lie on the same line then they are collinear ~true Biconditional: Points are collinear if and only if they lie on the same line.
biconditional
A single true statement that combines a true conditional and its true converse using the phrase "if and only if"
Thm. 2-4
All right angles are congruent
counterexample
An example that shows that a conjecture is incorrect. Example: Conjecture= If the name of a month starts with the letter J, it is a summer month. Counterexample= January starts with J and it's a winter month.
conditional
An if-then statement.
Transitive Property of Equality
If a = b and b = c, then a = c
Division Property of Equality
If a = b and c ≠ 0, then a/c = b/c
Multiplication Property of Equality
If a = b, then a * c = b * c
Addition Property of Equality
If a = b, then a + c = b + c
Subtraction Property of Equality
If a = b, then a - c = b - c
Symmetric Property of Equality
If a = b, then b = a
Substitution Property of Equality
If a = b, then b can replace a in any expression
Law of Detachment
If a conditional statement is true and its hypothesis is true, then its conclusion is true
Thm. 2-3: Congruent Complements Theorem
If two angles are complementary to the same angle (or of congruent angles, then the two angles are congruent
Thm. 2-5
If two angles are congruent and supplementary, then each is a right angle
Thm. 2-2: Congruent Supplements Theorem
If two angles are supplementary to the same angle or congruent angles, then the two angles are congruent
Transitive Property of Congruence
If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C
Symmetric Property of Congruence
If ∠A ≅ ∠B, then ∠B≅ ∠A
complementary angles
Two angles whose sum is 90 degrees
inverse
Negate both the hypothesis and the conclusion of the conditional.
converse
Switch the hypothesis and the conclusion in a conditional.
contrapositive
Switches the hypothesis and the conclusion and negates both.
Law of Syllogism
Takes two conditional statements and forms a conclusion by combining the hypothesis of one statement and the conclusion of another. Example: If a quadrilateral is a square, then it has four right angles. If a quadrilateral has four right angles, then it is a rectangle. Conclusion: If a quadrilateral is a square, then it is a rectangle.
negation
The negation of a statement is the opposite of the statement. . Example: Statement= The sky is blue. Negation = The sky is not blue.
hypothesis
The part of the conditional that comes after the word "if"
conclusion
The part of the conditional that comes after the word "then"
Reflexive Property of Congruence
∠A ≅ ∠A