Visual explanation of logical statements in geometry, showing how to derive converse, inverse, and contrapositive from an if-then statement.

You're asking for a solution to the problem presented in the image, which is:

> What if your sister told you, "If you do the dishes, then I will help you with your homework"? What's a statement that is *logically equivalent* to what your sister said?

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✔ Step-by-Step Solution

This is a logic problem about conditional statements and their related forms: converse, inverse, and contrapositive.

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🔹 1. Identify the original conditional statement

Your sister says:
> "If you do the dishes, then I will help you with your homework."

This is an if-then statement:
If p, then q

Let’s define:
- p = You do the dishes
- q = I will help you with your homework

So the original statement is:
> p → q

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🔹 2. Understand the related forms

From the lesson in the image, we know:

| Form | Symbolic Form | English Example (based on weather/car) |
|----------------|---------------|-----------------------------------------------|
| Original | p → q | If weather is nice, I’ll wash the car |
| Converse | q → p | If I wash the car, then weather is nice |
| Inverse | ~p → ~q | If weather is not nice, I won’t wash the car |
| Contrapositive | ~q → ~p | If I don’t wash the car, then weather isn’t nice |

> 💡 Key Point from the Lesson:
> The contrapositive is logically equivalent to the original statement. That means if the original is true, the contrapositive is also true — and vice versa.

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🔹 3. Apply this to your sister’s statement

Original:
> If you do the dishes, then I will help you with your homework.
> (p → q)

Contrapositive:
> If I do NOT help you with your homework, then you did NOT do the dishes.
> (~q → ~p)

✔ This is the logically equivalent statement.

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🧠 Why is this logically equivalent?

Think of it this way:
If your sister’s promise is true (“If you do the dishes, I’ll help”), then the *only way* she wouldn’t help is if you *didn’t* do the dishes.
That’s exactly what the contrapositive says.

It’s the same logical relationship — just stated backwards and negated.

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✘ Why not the converse or inverse?

- Converse: “If I help you with homework, then you did the dishes.”
→ Not necessarily true. Maybe she helps even if you didn’t do the dishes.
→ Not logically equivalent.

- Inverse: “If you didn’t do the dishes, then I won’t help.”
→ Also not necessarily true. She might still help out of kindness.
→ Not logically equivalent.

Only the contrapositive preserves the exact logical meaning.

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✔ Final Answer:

> “If I do not help you with your homework, then you did not do the dishes.”

This is the logically equivalent statement to your sister’s original promise.

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Let me know if you’d like to see truth tables or more examples!