22. (lse) if a quadrilateral does not have four right angles, then it is not a rectangle.
a. inverse
b. contrapositive
c. converse
d. either a or c
e. none of the above
23. select the correct proof from the options listed.
given: rectangle abcd; m is the midpoint of $\overline{ad}$

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Accepted Answer

# Brief Explanations:
First, recall the definitions:
- **Inverse** of a conditional statement $ p \to q $ is $
eg p \to
eg q $.
- **Contrapositive** is $
eg q \to
eg p $.
- **Converse** is $ q \to p $.

The original statement: "If a quadrilateral does not have four right angles ($
eg p $), then it is not a rectangle ($
eg q $)" (where $ p $: "has four right angles", $ q $: "is a rectangle"). Wait, no—wait, the standard definition: A rectangle ($ q $) has four right angles ($ p $), so original conditional is $ q \to p $? Wait, no, the given statement is: "If a quadrilateral does not have four right angles ($
eg p $), then it is not a rectangle ($
eg q $)". Wait, let's re-express: Let $ p $: "quadrilateral has four right angles", $ q $: "quadrilateral is a rectangle". Then the original statement is $
eg p \to
eg q $.

Now, the **contrapositive** of $
eg p \to
eg q $ would be $ q \to p $ (since contrapositive of $ A \to B $ is $
eg B \to
eg A $; here $ A =
eg p $, $ B =
eg q $, so contrapositive is $
eg (
eg q) \to
eg (
eg p) $, i.e., $ q \to p $). Wait, no—wait, maybe I mixed up. Let's start over.

Original conditional: "If (quadrilateral does not have four right angles), then (it is not a rectangle)". Let’s write it as $ \text{Not } p \to \text{Not } q $, where $ p $: "has four right angles", $ q $: "is a rectangle".

The **contrapositive** of $ A \to B $ is $
eg B \to
eg A $. So for $ A =
eg p $, $ B =
eg q $, contrapositive is $
eg (
eg q) \to
eg (
eg p) $, which simplifies to $ q \to p $ ("If it is a rectangle, then it has four right angles"), which is true. But wait, the given statement is $
eg p \to
eg q $. Wait, no—maybe the original statement is the contrapositive of the standard "If it is a rectangle, then it has four right angles" ($ q \to p $). The contrapositive of $ q \to p $ is $
eg p \to
eg q $, which matches the given statement. So the given statement is the contrapositive of $ q \to p $. Wait, but the options: Let's check the options again.

Wait, the question is about what the given statement is (Inverse, Contrapositive, Converse, etc.). Wait, maybe I misread. Let's re-express the original statement:

Let the standard definition: A rectangle (q) has four right angles (p), so $ q \to p $. The contrapositive of $ q \to p $ is $
eg p \to
eg q $ ("If not p (no four right angles), then not q (not a rectangle)"), which is exactly the given statement. So the given statement is the contrapositive of $ q \to p $. Wait, but the options: Option B is "Contrapositive". Wait, no—wait, maybe the original conditional is $ p \to q $, but here it's $
eg p \to
eg q $. Wait, no—let's confirm:

- Inverse of $ p \to q $ is $
eg p \to
eg q $.
- Contrapositive of $ p \to q $ is $
eg q \to
eg p $.
- Converse of $ p \to q $ is $ q \to p $.

Ah! Here's the key: If the original conditional were $ p \to q $ ("If a quadrilateral has four right angles, then it is a rectangle"), then its inverse would be $
eg p \to
eg q $ ("If a quadrilateral does not have four right angles, then it is not a rectangle"). Wait, that's the given statement! So the original conditional (implicit) is $ p \to q $ ("has four right angles → is a rectangle"), and the given statement is $
eg p \to
eg q $, which is the **inverse**? Wait, no—wait:

Wait, no—let's define $ p $ and $ q $ correctly. Let $ p $: "quadrilateral is a rectangle", $ q $: "quadrilateral has four right angles". Then the standard statement is $ p \to q $ ("If it is a rectangle, then it has four right angles").

- Inverse of $ p \to q $ is $
eg p \to
eg q $ ("If it is not a rectangle, then it does not have four right angles").
- Contrapositive of $ p \to q $ is $
eg q \to
eg p $ ("If it does not have four right angles, then it is not a rectangle").

Oh! Now I see. I mixed up $ p $ and $ q $. Let's redefine:

Let $ p $: "quadrilateral is a rectangle", $ q $: "quadrilateral has four right angles". Then:

- Original conditional: $ p \to q $ ("If rectangle, then four right angles").
- Contrapositive: $
eg q \to
eg p $ ("If no four right angles, then not a rectangle")—which is the given statement!

So the given statement is the contrapositive of $ p \to q $. Wait, but the options: Option B is "Contrapositive". Wait, but earlier confusion was due to misdefining $ p $ and $ q $. Let's confirm with definitions:

- **Contrapositive** of $ p \to q $ is $
eg q \to
eg p $.
- The given statement is "If (no four right angles, $
eg q $), then (not a rectangle, $
eg p $)"—which is $
eg q \to
eg p $, the contrapositive of $ p \to q $.

Wait, but the options: Let's check the options again. The question is: "If a quadrilateral does not have four right angles, then it is not a rectangle." What is this statement (Inverse, Contrapositive, etc.)?

Wait, let's list the options:

A. Inverse: $
eg p \to
eg q $ (where $ p $: "is a rectangle", $ q $: "has four right angles")—no, inverse of $ p \to q $ is $
eg p \to
eg q $.

B. Contrapositive: $
eg q \to
eg p $ (where $ p $: "is a rectangle", $ q $: "has four right angles")—yes, because contrapositive of $ p \to q $ is $
eg q \to
eg p $.

C. Converse: $ q \to p $ ("If has four right angles, then is a rectangle").

D. Either A or C: No.

E. None: No.

Wait, no—wait, the given statement is $
eg q \to
eg p $ (where $ p $: "is a rectangle", $ q $: "has four right angles"), which is the contrapositive of $ p \to q $. So the correct option is B. Contrapositive.

Wait, but earlier I thought the given statement was the inverse, but that was because I misassigned $ p $ and $ q $. The key is to define $ p $ as the hypothesis of the original (implicit) conditional. The implicit original conditional is "If a quadrilateral is a rectangle, then it has four right angles" ($ p \to q $, $ p $: "is a rectangle", $ q $: "has four right angles"). Then the contrapositive is $
eg q \to
eg p $ ("If no four right angles, then not a rectangle"), which is the given statement. So the given statement is the contrapositive.

# Answer:
B. Contrapositive

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