Let's go through each question on the worksheet step by step, solving and explaining each one clearly.
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1. A biconditional statement combines a conditional and its _________.
Answer: *converse*
Explanation:
A biconditional statement is formed when both a conditional and its converse are true. So, it combines a conditional ("if p, then q") and its
converse ("if q, then p").
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2. A biconditional statement can be written in the form "p if and only if q" which means "if p, then q, and if _________, then _________."
Answer: *q, p*
Explanation:
The phrase "p if and only if q" means:
- If p is true, then q is true (conditional),
and
- If q is true, then p is true (converse).
So:
"if p, then q, and if q, then p"
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3. Biconditional: A cat is happy if and only if it is purring.
Conditional: If a cat is happy, then it is purring.
Converse: ___________________________
Answer: *If a cat is purring, then it is happy.*
Explanation:
To find the
converse, switch the hypothesis and conclusion of the conditional.
Original conditional: If
happy, then
purring
Converse: If
purring, then
happy
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4. Biconditional: A figure is a segment if and only if it is straight and has two endpoints.
Conditional: If a figure is a segment, then it is straight and has two endpoints.
Converse: ___________________________
Answer: *If a figure is straight and has two endpoints, then it is a segment.*
Explanation:
Again, reverse the conditional:
Original: If
segment, then
straight and two endpoints
Converse: If
straight and two endpoints, then
segment
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5. Conditional: If two angles share a side, then they are adjacent.
Converse: If two angles are adjacent, then they share a side.
Biconditional: ___________________________
Answer: *Two angles are adjacent if and only if they share a side.*
Explanation:
Since both the conditional and converse are true (in geometry, this is accepted as a definition), we can combine them into a biconditional using "if and only if".
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6. Conditional: If your temperature is normal, then your temperature is 98.6°F.
Converse: If your temperature is 98.6°F, then your temperature is normal.
Biconditional: ___________________________
Answer: *Your temperature is normal if and only if it is 98.6°F.*
Explanation:
This assumes that 98.6°F is defined as "normal" temperature. Since both statements are considered equivalent in this context, we write the biconditional.
> Note: In reality, normal body temperature varies slightly, but for this problem, we treat it as a biconditional.
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7. The tea kettle is whistling if and only if the water is boiling.
Conditional: ___________________________
Converse: ___________________________
Answer:
-
Conditional: If the tea kettle is whistling, then the water is boiling.
-
Converse: If the water is boiling, then the tea kettle is whistling.
Explanation:
The biconditional "P if and only if Q" means:
- P → Q (conditional)
- Q → P (converse)
So:
- P = "tea kettle is whistling"
- Q = "water is boiling"
Thus:
- Conditional: If the tea kettle is whistling, then the water is boiling.
- Converse: If the water is boiling, then the tea kettle is whistling.
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✔ Final Answers:
1.
converse
2.
q, p
3.
If a cat is purring, then it is happy.
4.
If a figure is straight and has two endpoints, then it is a segment.
5.
Two angles are adjacent if and only if they share a side.
6.
Your temperature is normal if and only if it is 98.6°F.
7.
-
Conditional: If the tea kettle is whistling, then the water is boiling.
-
Converse: If the water is boiling, then the tea kettle is whistling.
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Parent Tip: Review the logic above to help your child master the concept of conditional statement worksheet.
