Problem 1 Identifying the Hypothesis and the Conclusion
What are the hypothesis and the conclusion of the conditional?
If an animal is a robin, then the animal is a bird.
Think
What would a Venn diagram look like?
A robin is a kind of bird, so the set of robins (R) should be inside the set of birds (B).
- Hypothesis (p): An animal is a robin.
- Conclusion (q): The animal is a bird.
✓ Got It?
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What are the hypothesis and the conclusion of the conditional?
If an angle measures 130, then the angle is obtuse.
Problem 2 Writing a Conditional
How can you write the following statement as a conditional?
Vertical angles share a vertex.
Think
Which part of the statement is the hypothesis (p)?
For two angles to be vertical, they must share a vertex. So the set of vertical angles (p) is inside the set of angles that share a vertex (q).
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Step 1 Identify the hypothesis and the conclusion.
Vertical angles share a vertex.
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Step 2 Write the conditional.
If two angles are vertical, then they share a vertex.
✓ Got It?
- How can you write “Dolphins are mammals” as a conditional?
The truth value of a conditional is either true or false. To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. A counterexample can help you determine whether a conditional with a true hypothesis is true. To show that the conditional is false, if you find one counterexample for which the hypothesis is true and the conclusion is false, then the truth value of the conditional is false.
Problem 3 Finding the Truth Value of a Conditional
Is the conditional true or false? If it is false, find a counterexample.
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If a woman is Hungarian, then she is European.
The conditional is true. Hungary is a European nation, so Hungarians are European.
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If a number is divisible by 3, then it is odd.
The conditional is false. The number 12 is divisible by 3, but it is not odd.
Plan
How do you find a counterexample?
Find an example where the hypothesis is true, but the conclusion is false. For part (B), find a number divisible by 3 that is not odd.
✓ Got It?
- Is the conditional true or false? If it is false, find a counterexample.
- If a month has 28 days, then it is February.
- If two angles form a linear pair, then they are supplementary.
Table of Contents
- 6-1 The Polygon Angle-Sum Theorems
- 6-2 Properties of Parallelograms
- 6-3 Proving That a Quadrilateral Is a Parallelogram
- 6-4 Properties of Rhombuses, Rectangles, and Squares
- 6-5 Conditions for Rhombuses, Rectangles, and Squares
- 6-6 Trapezoids and Kites
- 6-7 Polygons in the Coordinate Plane
- 6-8 and 6-9 Coordinate Geometry and Coordinate Proofs