Geometry Ch 2 - Reasoning and Proof Flashcards
What are the parts of a conditional statement?
There are two parts to every conditional statement:
1) The part following if is the hypothesis, and
2) The part following then is the conclusion.
How do you prove a conditional is either true or false?
To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. To show that a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.
What is the converse of a conditional?
The converse of a conditional switches the hypothesis and the conclusion.
Example: Writing the Converse of a Conditional
Write the converse of the following conditional:
Condilional
If two lines intersect to form right angles, then they are perpendicular.
Converse
If two lines are perpcndicular, then they intersect to form right angles.
What is a Biconditional Statement?
In logic, a biconditional is a compound statement formed by combining two conditionals under “and.” Biconditionals are true when both statements (facts) have the exact same truth value.
A biconditional is read as “[some fact] if and only if [another fact]” and is true when the truth values of both facts are exactly the same:
In other words, it is a valid biconditional statement when if and only if BOTH ARE TRUE or BOTH ARE FALSE.
Example: Writing a Biconditional
Consider this true conditional statement. Write its converse. If the converse is also
true, combine the statements as a biconditional.
Conditional
If two angles have the same measure, then the angles arc congruent.
Converse
If two angles are congruent, then the angles have the same measure.
The converse is also true.
Since both the conditional and its converse are true, you can combine them in a true biconditional by using the phrase if and only if:
Biconditional
Two angles have the same measure if and only if the angles are congruent.
What is Deductive (or logical) Reasoning?
The process of reasoning logically from given statemetns to a conclusion. If the given statements are true, deductive reasoning produces a true conclusion.
What is the Law of Detachment?
If a conditional is true and its hypthesis is true, then its conclusion is true. In symbolic form: If p → q is a true statement and p is true, then q is true.
What is the Law of Syllogism?
Another law of deductive reasoning that allows you to state a conclusion from two conditional statements when the conclusion of one statement is the hypothesis of the other statement.
In other words: If p → q and q → r are true, then p → r is a true statement.
What is the Addition Property of Equality?
If a = b, then a + c = b +c
What is the Subtraction Property of Equality?
If a = b, then a - c = b - c
What is the Multiplication Property of Equality?
If a = b, then a · c = b · c
What is the Division Property of Equality?
If a = b and c ≠ 0, then a/c = b/c
What is the Reflexive Property of Equality?
a = a
What is the Symmetric Property of Equality?
If a = b, then b = a
What is the Transitive Property of Equality?
If a = b and b = c, then a = c
What is the Substitution Property of Equality?
If a = b, then b can replace a in any expression.
What is the Reflexive Property of Congruence?
What is the Symmetric Property of Congruence?
What is the Transitive Property of Congruence?
What is the Vertical Angles Theorum?
