Geometry · Unit 1
Write if-and-only-if statements, and turn a definition into a biconditional.
If and only if
p ↔ q means both directions: p → q and q → p.
Definitions
A good definition is a biconditional. Both directions are true.
Example: A polygon is a hexagon if and only if it has exactly six sides.
In words
Say “p if and only if q,” or write both if-then sentences.
Not the same as one arrow
p → q is one direction. p ↔ q needs both.
Join two true directions with if and only if.
p ↔ q
A triangle is equilateral if and only if it has three congruent sides.
Answer: A triangle is equilateral iff it has three congruent sides. p ↔ q
Two lines are perpendicular if and only if they form a right angle.
Answer: Two lines are perpendicular if and only if they form a right angle.
p ↔ q
Answer: p ↔ q
If a figure is a square, then it is a rhombus with a right angle.
If a figure is a rhombus with a right angle, then it is a square.
Answer: both directions
↔ is if and only if.
→ is one-way if-then.
Answer: ↔
Today is Saturday if and only if tomorrow is Sunday.
Answer: Today is Saturday if and only if tomorrow is Sunday.
Turn a definition into a biconditional.
A biconditional can be written either way.
A point is the midpoint of a segment if and only if it divides the segment into two congruent segments.
A point divides a segment into two congruent segments if and only if it is the midpoint of the segment.
Answer: A point is the midpoint of a segment if and only if it divides the segment into two congruent segments. Either order is correct.
A biconditional can be written either way.
An angle is a right angle if and only if its measure is 90°.
An angle has measure 90° if and only if it is a right angle.
Answer: An angle is a right angle if and only if its measure is 90°. Either order is correct.
A biconditional can be written either way.
Two coplanar lines are parallel if and only if they do not intersect.
Two coplanar lines do not intersect if and only if they are parallel.
Answer: Two coplanar lines are parallel if and only if they do not intersect. Either order is correct.
A biconditional can be written either way.
Two angles form a linear pair if and only if they are adjacent and their noncommon sides form a line.
Two angles are adjacent and their noncommon sides form a line if and only if they form a linear pair.
Answer: Two angles form a linear pair if and only if they are adjacent and their noncommon sides form a line. Either order is correct.
If a triangle is isosceles, then it has at least two congruent sides.
If a triangle has at least two congruent sides, then it is isosceles.
A biconditional can be written either way.
A triangle is isosceles if and only if it has at least two congruent sides.
A triangle has at least two congruent sides if and only if it is isosceles.
Answer: p → q and q → p. Either order of the biconditional is correct.
p ↔ q
A biconditional can be written either way.
A polygon is a triangle if and only if it has exactly three sides.
A polygon has exactly three sides if and only if it is a triangle.
Answer: A polygon is a triangle if and only if it has exactly three sides. Either order is correct.
Both directions, and tell a biconditional from a one-way conditional.
Two angles are complementary if and only if the sum of their measures is 90°.
If two angles are complementary, then the sum of their measures is 90°.
If the sum of two angles’ measures is 90°, then the angles are complementary.
Answer: complementary iff measures sum to 90°
If and only if is the biconditional.
B
Answer: B
The converse is: If a figure is a rectangle, then it is a square.
That converse is false.
No. Both directions are not true.
Answer: No
p ↔ q
A whole number is divisible by 6 if and only if it is divisible by 2 and by 3.
Answer: divisible by 6 iff divisible by 2 and by 3
A line, ray, or segment is a bisector of a segment if and only if it intersects the segment at its midpoint.
Answer: bisector iff it meets the segment at its midpoint
If an integer is a multiple of 10, then it ends in 0.
If an integer ends in 0, then it is a multiple of 10.
An integer is a multiple of 10 if and only if it ends in 0.
Answer: multiple of 10 iff it ends in 0
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