If-Then
In the conditional statement, "If an angle measures 90 degrees, then it is a right angle," what is the hypothesis?
An angle measures 90 degrees.
If a conditional statement is represented as p ->q, what is the symbolic representation of its converse?
q -> p
True or False: A conditional statement and its contrapositive always share the exact same truth value.
True
Decompose this biconditional midpoint definition into its two underlying conditional statements: "M is the midpoint of segment AB if and only if AM = MB (and M is between A and B)."
Decompose this biconditional midpoint definition into its two underlying conditional statements: "M is the midpoint of segment AB if and only if AM = MB (and M is between A and B)."
In the conditional statement, "If two angles form a linear pair, then they are supplementary," what is the conclusion?
They are supplementary.
Write the inverse of the statement: "If it is Monday, then I am at school."
"If it is not Monday, then I am not at school."
Provide a counterexample to disprove this statement: "All congruent angles are vertical angles."
Any two angles that measure the same but are not opposite each other
True or False: If the converse of a true conditional statement is proven to be false, then its inverse must be true.
Rewrite the following definition in "If... then" form: "Vertical angles are congruent."
If two angles are vertical angles, then they are congruent.
Write the converse of the statement: "If a polygon is an octagon, then it has eight sides."
"If a polygon has eight sides, then it is an octagon
Provide a counterexample to disprove this statement: "All segment bisectors are perpendicular bisectors."
A line segment bisected by a line that crosses it at a non-right angle (such as a 45 degree angle)
In a formal two-column proof, if you write the statement: m<angle 1 = m<angle 2$" because you established in the previous line that "<angle 1 = <angle 2", what is the reason you must state?
"Definition of congruent angles."
What are the starting rules of geometry that we accept as true without needing a proof?
Postulates (or Axioms).
Write the contrapositive of this statement: "If two angles are vertical angles, then they are congruent."
"If two angles are not congruent, then they are not vertical angles."
Judge the validity of this argument: "All squares are rectangles. Quadrilateral ABCD is not a square. Therefore, ABCD is not a rectangle."
Invalid (ABCD could be a rectangle that is not a square, such as a 3x5 rectangle).
A student states: "Since the converse of 'If two angles are vertical, then they are congruent' is false, it is impossible for congruent angles to be vertical." What logical error has the student made?
The student has confused "not always true" with "never true." The converse being false simply means congruent angles are not necessarily vertical, but they still can be.
Translate the following conditional statement into its common symbolic logic notation: "If p, then q."
p -> q
A mathematically precise definition can be written as a biconditional statement, which uses what specific four-word phrase?
if and only if
Judge the validity of this argument: "If you are a student at Clay County Schools, then you are a student at Fleming Island High. You are not a Fleming Island High student. Therefore, you are not a Clay County Schools student."
Valid
A student presents a proof showing that a line segment has been divided into two congruent segments, and concludes: "Therefore, this line is the perpendicular bisector." Explain why this is invalid and state how to correct the argument.
It is invalid because the student only proved a segment bisector, not a perpendicular one. To make it valid, they must also prove that the intersecting line forms a right angle (90 degree) with the segment