BICONDITIONALS AND DEFINITIONS

BIG IDEA: REASONING AND PROOF

ESSENTIAL UNDERSTANDINGS:• A definition is good if it can be written as a

biconditional.• Every biconditional can be written as two

conditionals that are converses of each other.

MATHEMATICAL PRACTICE:

Attend to precision

GETTING READY Look at the examples of insects and

noninsects below. How would you complete the following sentence: “If an animal is an insect, then . . . ? Explain your reasoning.

MATH BACKGROUND

The first terms learned in Geometry must be understood and undefined. The terms that follow can be explained using only undefined terms or a combination of undefined terms and previously defined terms.

Postulates are assumed to be true without proof, and theorems are proved using postulates and previously proved theorems.

DEFINITIONS

All definitions can be interpreted “forward” and “backward”.

Vertical Angles: Two angles whose sides form two pairs of opposite rays. Two pairs of opposite rays form the sides of two angles.

Angle Bisector: A ray that divides an angle into two adjacent angles that are

congruent. Two adjacent angles that are congruent in an angle divided by

a ray.

Perpendicular Lines: Two lines that intersect to form a right angle. Two lines that form a right angle are perpendicular.

EX 1: USE THE DIAGRAM TO DETERMINE WHETHER THE STATEMENT IS TRUE OR FALSE

a) Points O, P and Q are collinear.

b) are supplementary

c) is perpendicular to

d) are complementary

e) Point Q is between point O and point P

f) are vertical angles

and MPO NQP

MP;<<<<<<<<<<<<< <

NQ;<<<<<<<<<<<<< <

and MPO MPQ

and LMO KMN

BICONDITIONAL STATEMENT

The combination of a _________________________ statement and its ____________________. A biconditional contains the words “ _________ and ____________ __________”.

EX: A point is a midpoint if and only if it divides a segment into two congruent segments.

If a point is a midpoint, then it divides a segment into two congruent segments

If a point divides a segment into two congruent segments, then it is a midpoint.

and as p q q p p q

EX 2: REWRITE THE BICONDITIONAL STATEMENT AS A CONDITIONAL AND ITS CONVERSE STATEMENTS.

a) Two lines intersect if and only if their intersection is exactly one point.

b) Two angles are supplementary if and only if the sum of the measures of the two angles is .

180

EX 3: THE FOLLOWING STATEMENT IS TRUE. WRITE THE CONVERSE AND DECIDE WHETHER IT IS TRUE OR FALSE. IF THE CONVERSE IS TRUE, COMBINE TO FORM A BICONDITIONAL

a)

b)

2If 49, then 7.x x

2If 4, then 2 or 2.x x x

GOOD DEFINITION A good definition is a statement that can help you

identify or classify an object. A good definition has several important components. It uses clearly understood terms. These terms should be

commonly understood or already defined. It is precise. Good definitions avoid words such as large, sort

of, and almost. It is reversible. That means you can write a good definition

as a true biconditional.

Which of the following are good definitions?

a) A fish is an animal that swims.b) Giraffes are animals with very long necks.c) A square is a figure with four right anglesd) A penny is a coin worth one cent.e) Perpendicular lines are two lines that intersect to form right

angles.

2.2 P. 8214 – 42 EVENS, 51 – 54 ALL, 5620 questions