Geometry (A): READ: Conditional Statement Reading

Conditional Statements

Learning Objectives

Recognize if-then statements.
Identify the hypothesis and conclusion of an if-then statement.
Write the converse, inverse, and contrapositive of an if-then statement.
Understand a biconditional statement.

Introduction

In geometry we reason from known facts and relationships to create new ones. You saw earlier that inductive reasoning can help, but it does not prove anything. For that we need another kind of reasoning. Now you will begin to learn about deductive reasoning, the kind of reasoning used throughout mathematics and science.

If-Then Statements

In geometry, and in ordinary life, we often make conditional, or if-then, statements.

Statement 1: If the weather is nice, I’ll wash the car. (“Then” is implied even if not stated.)
Statement 2: If you work overtime, then you’ll be paid time-and-a-half.
Statement 3: If $2$ divides evenly into $x$ , then $x$ is an even number.
Statement 4: If a triangle has three congruent sides, it is an equilateral triangle. (“Then” is implied; this is a definition.)
Statement 5: All equiangular triangles are equilateral. (“If” and “then” are both implied.)

An if-then statement has two parts.

The “if” part is called the hypothesis.
The “then” part is called the conclusion.

For example, in statement 2 above, the hypothesis is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.”

Look at statement 1 above. Even though the word “then” is not actually present, the statement could be rewritten as: If the weather is nice, then I’ll wash the car. This is the meaning of statement 1. The hypothesis is “the weather is nice.” The conclusion is “I’ll wash the car.”

Statement 5 is a little more complicated. “If” and “then” are both implied without being stated. Statement 5 can be rewritten as: If a triangle is equiangular, then it is equilateral.

What is meant by an if-then statement? Suppose your friend makes the statement in statement 2 above, and adds another fact.

If you work overtime, then you’ll be paid time-and-a-half.
You worked overtime this week.

If we accept these statements, what other fact must be true? Combining these two statements, we can state with no doubt:

You’ll be paid time-and-a-half this week.

Let’s analyze statement 1, which was rewritten as: If the weather is nice, then I’ll wash the car. Suppose we accept statement 1 and another fact: I’ll wash the car.

Can we conclude anything further from these two statements? No. Even if the weather is not nice, I might wash the car. We do know that if the weather is nice I’ll wash the car. We don’t know whether or not I might wash the car even if the weather is not nice.

Converse, Inverse, and Contrapositive of an If-Then Statement

Look at statement 1 above again.

If the weather is nice, then I’ll wash the car.

This can be represented in a diagram as:

If $p$ then $q$ .

$p = \text{the weather is nice}& &q = \text{I'll wash the car}$

“If $p$ then $q$ ” is also written as

$p \rightarrow q$

Notice that conditional statements, hypotheses, and conclusions may be true or false. $p,$ $q,$ and the statement “If $p,$ then $q$ ” may be true or false.

In deductive reasoning we sometimes study statements related to a given if-then statement. These are formed by using $p,$ $q,$ and their opposites, or negations (“not”). Note that “not $p$ ” is written in symbols as $\lnot p$ .

$p,q,$ $\lnot p,$ and $\lnot q$ can be combined to produce new if-then statements.

The converse of $p \rightarrow q$ is $q \rightarrow p.$
The inverse of $p \rightarrow q$ is $\lnot p \rightarrow \lnot q.$
The contrapositive of $p \rightarrow q$ is $\lnot q \rightarrow \lnot p.$

Now let’s go back to statement 1: If the weather is nice, then I’ll wash the car.

$p \rightarrow q& &p & = \text{the weather is nice}\\ & &q &= \text{I’ll wash the car}\\ &&\lnot p &= \text{the weather is not nice}\\ &&q &= \text{I’ll wash the car (or I wash the car)}\\ &&\lnot q &= \text{I won’t wash the car (or I don’t wash the car)}$

Converse: $q \rightarrow p\;$ If I wash the car, then the weather was nice.

Inverse: $\lnot p \rightarrow \lnot q\;$ If the weather is not nice, then I won’t wash the car.

Contrapositive: $\lnot q \rightarrow \lnot p\;$ If I don’t wash the car, then the weather is not nice.

Notice that if we accept statement 1 as true, then the converse and inverse may, or may not, be true. But the contrapositive is true. Another way to say this is: The contrapositive is logically equivalent to the original if-then statement. In future work you may be asked to prove an if-then statement. If it’s easier to prove the contrapositive, then you can do this since the statement and its contrapositive are equivalent.

Example 1

Statement:

If it is February, then there are only 28 days in the month.

Converse:

If there are only 28 days in the month, then it is February.

A counterexample is February in the year 2008. Because 20085 is a leap year, the month of February has 29 days.

Inverse:

If there is not only 28 days in the month, then it is not February.

Contrapositive:

If it is not February, then there is not only 28 days in the month.

Example 2

Statement: If $AB = BC,$ then $B$ is the midpoint of $AC$ . False (as shown below).

Needs $AB=BC$

Converse: If $B$ is the midpoint of $\overline{AC}$ , then $AB = BC$ . True.

Inverse: If $AB \neq BC$ , then $B$ is not the midpoint of $\overline{AC}$ . True.

Contrapositive: If $B$ is not the midpoint of $\overline{AC}$ , then $AB \neq BC$ False (see the diagram above).

Biconditional Statements

You recall that the converse of “If $p$ then $q$ ” is “If $q$ then $p$ .” When these two are combined, we have a biconditional statement.

Biconditional: $p\ \rightarrow\ q$ and $q\ \rightarrow\ p$

In symbols, this is written as: $p\ \leftrightarrow\ q$

We read $p\ \leftrightarrow\ q$ as: “ $p$ if and only if $q$ ”

Example 3

True statement: $m\angle{ABC} > 90^\circ$ if and only if $\angle{ABC}$ is an obtuse angle.

You can break this down to say:

If $m\angle{ABC} > 90^\circ$ then $\angle{ABC}$ is an obtuse angle and if $\angle{ABC}$ is an obtuse angle then $m\angle{ABC} > 90^\circ$ .

Notice that both parts of this biconditional are true; the biconditional itself is true.

You most likely recognize this as the definition of an obtuse angle.

Geometric definitions are biconditional statements that are true.

Example 4

Let $p$ be "if two lines are perpendicular"

Let $q$ be "then they intersect to form right angles"

a. Is $p$ $\rightarrow$ $q$ true?

Yes.

$p\ \rightarrow\ q$ is "If two lines are perpendicular, then they intersect to form right angles".

You will be learning this year that this statement is true. The definition of perpendicular lines is that they intersect to form four right angles.

b. Is $q \ \rightarrow\ p$ true?

Yes.

$q\ \rightarrow\ p$ is "If two lines intersect to form right angles, then they are perpendicular".

c. Is $p \leftrightarrow q$ true?

Yes.

$p\ \leftrightarrow\ q$ is "Two lines are perpendicular if and only if they intersect to form four right angles".

Once again notice that both parts of this biconditional are true; the biconditional itself is true.

Example 5

Let $p$ be "If x=3 "

Let $q$ be "then $$\left|x\right|$$=3"

a. Is $p$ $\rightarrow$ $q$ true?

Yes.

$p\ \rightarrow\ q$ is if x=3 then $$\left|x\right|$$=3

b. Is $q \ \rightarrow\ p$ true?

No.

$q\ \rightarrow\ p$ is if $$\left|x\right|$$=3 then x=3

From algebra we know that if $$\left|x\right|$$=3 then x=3 or x=-3.

c. Is $p \leftrightarrow q$ true?

No.

$p\ \leftrightarrow\ q$ is x=3 if and only if $$\left|x\right|$$=3

This statement is false.

Note that if either $p\ \rightarrow\ q$ or $q\ \rightarrow\ p$ is false, then $p\ \leftrightarrow\ q$ is false.

Lesson Summary

In this lesson you have learned how to express mathematical and other statements in if-then form. You also learned that each if-then statement is linked to variations on the basic theme of “If $p$ then $q$ .” These variations are the converse, inverse, and contrapositive of the if-then statement. Biconditional statements combine the statement and its converse into a single “if and only if” statement. Definitions are an important type of biconditional, or if-and-only-if, statement.

Points to Consider

We called points, lines, and planes the building blocks of geometry. We will soon see that hypothesis, conclusion, as well as if-then and if-and-only-if statements are the building blocks that deductive reasoning, or logic, is built on. This type of reasoning will be used throughout your study of geometry. In fact, once you understand logical reasoning you will find that you apply it to other studies and to information you encounter all your life.

Last modified: Monday, June 28, 2010, 12:19 PM