chapter 2: reasoning & proof 2.2 biconditionals & definitions
TRANSCRIPT
Chapter 2: Reasoning & Proof
2.2
Biconditionals & Definitions
Biconditionals
• when a conditional and its converse are true, you can combine them as a true biconditional
• connects the conditional and its converse with the word “and”
• written shorter by using “if and only if”
Example 1
• Write the converse. If the converse is also true, combine the statements as a biconditional:
• If two angles have the same measure, then the angles are congruent.
Quick Check 1
• Write the converse. If the converse is also true, combine the statements as a biconditional:
• If three points are collinear, then they lie on the same line.
Example 2
• Write two statements that form the biconditional:
• A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
Quick Check 2
• Write two statements that form this biconditional:
• A number is prime if and only if it has only two distinct factors, 1 and itself.
Summary
• biconditional statements:
• p↔q
• p if and only if q
Definitions
• A good definition:
• statement that help you identify or classify an object
• uses clearly understood terms
• precise
• reversible
Example 3
• Show that this definition of perpendicular lines is reversible. Then, write it as a true biconditional:
• Perpendicular lines are two lines that intersect to form right angles.
Quick Check 3
• Show that the definition of right angle is reversible. Then, write it as a true biconditional:
• A right angle is an angle whose measure is 90.