1.14: Conditional Statements
Learning Objectives
- Write definitions as “if-then” statements.
- Identify the hypothesis and conclusion of an if-then statement.
- Write a counterexample to disprove a conjecture.
Conditional Statements, or If-Then Statements
In geometry, and in ordinary life, we often make conditional, or if-then, statements.
- Statement 1: If 2 divides evenly into \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} is an even number.
- Statement 2: If the weather is nice, I will wash the car. (“Then” is implied even if not stated.)
- Statement 3: If a triangle has three congruent sides, it is an equilateral triangle. (“Then” is implied; this is a definition.)
- Statement 4: All equiangular triangles are equilateral. (“If” and “then” are both implied.)
A conditional statement and an _______-_______ statement are the same thing.
Hypothesis and Conclusions
An if-then statement has two parts:
- The “if” part is called the hypothesis.
- The “then” part is called the conclusion.
The “if” part of an “if-then” statement is called the _____________________________.
The “then” part of an “if-then” statement is called the _____________________________.
For example, in statement 1 above:
The hypothesis is “2 divides evenly into \begin{align*}x\end{align*}.”
The conclusion is “\begin{align*}x\end{align*} is an even number.”
What does it mean to imply something?
What does it mean if something is implied?
To “imply” is to say something without saying it directly.
If something is implied, it isn’t said directly, but we assume it is there. The meaning is suggested, but it isn’t actually there.
- To “imply” means to state something without stating it ________________________.
Look at statement 2 from the previous page:
Even though the word “then” is not actually present, the statement could be rewritten as:
If the weather is nice, then I will wash the car.
This is the meaning of statement 2. The word “then” is implied.
The hypothesis is “the weather is nice.” The conclusion is “I will wash the car.”
Statement 4 is a little more complicated:
- Statement 4: All equiangular triangles are equilateral. (“If” and “then” are both implied.)
“If” and “then” are both implied without being stated.
Statement 4 can be rewritten as: If a triangle is equiangular, then it is equilateral.
The hypothesis is: ___________________________________________________________
The conclusion is: ___________________________________________________________
Reading Check:
Identify the hypothesis and conclusion in each of these statements.
1. If a polygon has three sides, then it is a triangle.
- Hypothesis:_____________________________________________________
______________________________________________________________
- Conclusion:_____________________________________________________
______________________________________________________________
2. A segment has two endpoints.
(\begin{align*}*\end{align*}Hint: First, change this into an if-then statement: If it is a ___________________, then it is a ____________________.)
- Hypothesis:_____________________________________________________
______________________________________________________________
- Conclusion:_____________________________________________________
______________________________________________________________
What is meant by an if-then statement? Suppose your friend makes the following statements:
- If an employee works overtime, then he or she will 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 will be paid time-and-a-half this week.
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.
- Deductive reasoning is also know as ______________________.
Conjectures and Counterexamples
A conjecture is an “educated guess” that is often based on examples in a pattern. Examples suggest a relationship, which can be stated as a possible rule, or conjecture, for the pattern. Numerous examples may make you strongly believe the conjecture. However, no number of examples can prove the conjecture. It is always possible that the next example would show that the conjecture does not work.
- A conjecture is an educated ____________________ based on examples in a pattern.
Example 1
Here are three numbers in a sequence:
1, 2, 3
What do you think the next number in the sequence will be?
Your answer is a conjecture. It is an educated guess. You do not know for sure what the next number will be.
Here is the same sequence with a few more numbers included:
1, 2, 3, 5, 8, 13...
If your first conjecture was that the next number would be 4, then your conjecture was wrong. The numbers in this sequence are found by adding the two previous numbers:
\begin{align*}1 + 2 &= 3\\ 2 + 3 & = 5\\ 3 + 5 & = 8\\ 5 + 8 &= 13\ \text{and so on} \ldots\end{align*}
Reading Check:
Ramona studied positive even numbers. She broke some positive even numbers down as follows:
\begin{align*}\mathbf{8} = 3 + 5 && \mathbf{14} = 5 + 9 && \mathbf{36} = 17 + 19 && \mathbf{82} = 39 + 43\end{align*}
What conjecture might be suggested by Ramona’s results?
Ramona made this conjecture:
Every positive even number is the sum of two different positive odd numbers.
If you can find just one example that makes this conjecture false, then you have disproven the entire statement. A statement that makes the conjecture false is called a counterexample. The prefix “counter” means “opposite” or “against.”
A counterexample is an example that goes “against” your conjecture.
Can you think of a counterexample to this statement? “All shapes have straight sides.”
A counterexample is an example that goes __________________________ your conjecture.
In other words, a counterexample is an example that proves your conjecture is false.
Reading Check:
Arthur is making figures for a graphic art project. He drew polygons and some of their diagonals:
Based on these examples, Arthur made this conjecture:
If a convex polygon has \begin{align*}n\end{align*} sides, then there are \begin{align*}n - 3\end{align*} diagonals from any given vertex of the polygon.
1. Is Arthur’s conjecture correct?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Can you find a counterexample to the conjecture?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw \begin{align*}n – 3\end{align*} diagonals if the polygon has \begin{align*}n\end{align*} sides.
Notice that we have not proved Arthur’s conjecture. Many examples have (almost) convinced us that it is true.
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