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.