Intro to Reasoning: Conditional Statements & Logic
A conditional statement is an if-then statement: “If two angles are vertical, then they are congruent.” Almost every theorem in geometry is written this way, so learning to take a conditional apart is the first real skill of proof writing.
Every conditional has two parts. The part after “if” is the hypothesis — the condition being assumed. The part after “then” is the conclusion — what must follow. Once you can name those two parts on sight, converses and counterexamples come quickly.
Hypothesis and conclusion
The hypothesis is everything after “if,” and the conclusion is everything after “then.” In “If two angles form a linear pair, then they are supplementary,” the hypothesis is “two angles form a linear pair” and the conclusion is “they are supplementary.” The words “if” and “then” themselves are not part of either piece.
Some statements hide the if-then form. “All squares have four sides” can be rewritten as “If a figure is a square, then it has four sides.” Rewriting a statement in if-then form is often the first step to analyzing it. The table below splits that rewritten statement into its two parts.
The converse
The converse of a conditional swaps the hypothesis and conclusion. The converse of “If a figure is a square, then it has four sides” is “If a figure has four sides, then it is a square.”
Here is the trap: a statement and its converse are not always both true. The original statement about squares is true, but the converse is false — a rectangle has four sides without being a square. Always test a converse on its own; never assume it inherits the truth of the original.
Counterexamples
A counterexample is one case where the hypothesis is true but the conclusion is false. One counterexample is enough to prove a conditional false.
To disprove “If a figure has four sides, then it is a square,” point to a rectangle that is not a square: the hypothesis holds (four sides) and the conclusion fails (not a square). Done — the statement is false.
Worked examples
Example 1: identify the parts
Name the hypothesis and conclusion of “If it is raining, then the ground is wet.”
Answer: Hypothesis: it is raining. Conclusion: the ground is wet.
Example 2: write and test the converse
Write the converse of “If two angles are vertical, then they are congruent.” Is the converse true?
Answer: Converse: “If two angles are congruent, then they are vertical.” It is false.
Example 3: find a counterexample
Find a counterexample to “All prime numbers are odd.”
Answer: The number is a counterexample.
Try one yourself
Common questions
Is the hypothesis always written first?
No. “The ground is wet if it is raining” puts the hypothesis last. Find the “if” — whatever follows it is the hypothesis, wherever it sits in the sentence.
If a conditional is true, is its converse true too?
Not necessarily. The converse is a separate statement that must be tested on its own. “If a figure is a square, then it has four sides” is true, but its converse is false.
How many counterexamples do I need?
One. A single case where the hypothesis is true and the conclusion is false makes the whole conditional false.
What if a statement has no “if” or “then” in it?
Rewrite it. “All right angles are congruent” becomes “If two angles are right angles, then they are congruent.” The subject of the claim becomes the hypothesis.
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