Question

conditional: \if two angles have a sum of 180 degrees, then they are supplementary.\ find the converse, inverse, and contrapositive. if two angles are supplementary, then they have a sum of 180 degrees. dropdown: inverse if two angles are not supplementary, then they do not have a sum of 180 degrees. dropdown: contrapositive if two angles do not have a sum of 180 degrees, then they are not supplementary. dropdown: inverse

Explanation:

Let the original conditional statement be \( p \to q \), where \( p \): "two angles have a sum of 180 degrees" and \( q \): "they are supplementary".

Step 1: Recall Definitions

• Converse: \( q \to p \) (swap hypothesis and conclusion)
• Inverse: \(

eg p \to
eg q \) (negate both hypothesis and conclusion)

• Contrapositive: \(

eg q \to
eg p \) (negate and swap hypothesis and conclusion)

Step 2: Analyze Each Statement

1. For "If two angles are supplementary, then they have a sum of 180 degrees":

• Here, we have \( q \to p \), which matches the definition of the converse. So the first dropdown should be "Converse".

1. For "If two angles are not supplementary, then they do not have a sum of 180 degrees":

• Here, we have \(

eg q \to
eg p \), which matches the definition of the contrapositive. Wait, no—wait, \(
eg q \to
eg p \) is contrapositive, but let's check again. Wait, the original \( p \) is "sum 180", \( q \) is "supplementary". So \(
eg q \) is "not supplementary", \(
eg p \) is "not sum 180". So \(
eg q \to
eg p \) is contrapositive. But wait, the second statement: "If two angles are not supplementary, then they do not have a sum of 180 degrees" is \(
eg q \to
eg p \), which is contrapositive? Wait no, wait inverse is \(
eg p \to
eg q \), contrapositive is \(
eg q \to
eg p \). Wait, let's re - express:

Original: \( p \) (sum 180) \( \to q \) (supplementary)

• Converse: \( q \to p \) (supplementary \( \to \) sum 180)
• Inverse: \(

eg p \to
eg q \) (not sum 180 \( \to \) not supplementary)

• Contrapositive: \(

eg q \to
eg p \) (not supplementary \( \to \) not sum 180)

So:

• The first statement: "If two angles are supplementary, then they have a sum of 180 degrees" is \( q \to p \), so it's the converse.
• The second statement: "If two angles are not supplementary, then they do not have a sum of 180 degrees" is \(

eg q \to
eg p \), so it's the contrapositive? Wait no, wait the second statement's structure: "not supplementary" ( \(
eg q \)) \( \to \) "not sum 180" ( \(
eg p \))—that is contrapositive.

• The third statement: "If two angles do not have a sum of 180 degrees, then they are not supplementary" is \(

eg p \to
eg q \), so it's the inverse? Wait no, \(
eg p \to
eg q \) is inverse. Wait, let's correct:

1. First statement: "If two angles are supplementary, then they have a sum of 180 degrees" — \( q \to p \) → Converse.
2. Second statement: "If two angles are not supplementary, then they do not have a sum of 180 degrees" — \(

eg q \to
eg p \) → Contrapositive? Wait no, contrapositive is \(
eg q \to
eg p \), yes. Wait, but the third statement: "If two angles do not have a sum of 180 degrees, then they are not supplementary" — \(
eg p \to
eg q \) → Inverse.

Wait, let's re - assign:

• Converse: \( q \to p \) → "If two angles are supplementary, then they have a sum of 180 degrees"
• Inverse: \(

eg p \to
eg q \) → "If two angles do not have a sum of 180 degrees, then they are not supplementary"

• Contrapositive: \(

eg q \to
eg p \) → "If two angles are not supplementary, then they do not have a sum of 180 degrees"

So the correct dropdowns are:

1. For "If two angles are supplementary, then they have a sum of 180 degrees" → Converse
2. For "If two angles are not supplementary, then they do not have a sum of 180 degrees" → Contrapositive
3. For "If two angles do not have a sum of 180 degrees, then they are not supplementary" → Inverse

Answer:

1. First dropdown: Converse
2. Second dropdown: Contrapositive
3. Third dropdown: Inverse