Question

6 given: if two angles form a linear pair, then they are supplementary.
given: if two angles are supplementary, then the sum of their measures is 180°.
conclusion: if two angles form a linear pair, then the sum of their measures is 180°.
answer:
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7 given: if two angles measure 60°, then they are congruent.
given: ∠j ≅ ∠k
conclusion: m∠j = 60° and m∠k = 60°
answer:
8 given: if kaylee gets at least a 95 on her final exam, she will get an a in geometry.
given: kaylee got a 98 on her final exam.
conclusion: kaylee got an a in geometry.
answer:
9 given: if it is raining, then you must bring an umbrella.
given: you bring your umbrella.
conclusion: it is raining.
answer:

Explanation:

Step1: Analyze the reasoning for each problem

• Problem 6:
• The first statement is \(p

ightarrow q\) (if two angles form a linear pair (\(p\)), then they are supplementary (\(q\))).

• The second statement is \(q

ightarrow r\) (if two angles are supplementary (\(q\)), then the sum of their measures is \(180^{\circ}\) (\(r\))).

• By the law of syllogism (\((p

ightarrow q)\land(q
ightarrow r)\Rightarrow(p
ightarrow r)\)), the conclusion is valid.

• Problem 7:
• The first statement is \(p

ightarrow q\) (if two angles measure \(60^{\circ}\) (\(p\)), then they are congruent (\(q\))).

• The second statement is \(r\) (\(\angle J\cong\angle K\)).
• There is no valid logical rule (like law of detachment (\(p

ightarrow q\) and \(p\) then \(q\)) or law of syllogism) to get from \(p
ightarrow q\) and \(r\) to \(p\) (i.e., \(m\angle J = 60^{\circ}\) and \(m\angle K=60^{\circ}\)). The conclusion is invalid.

• Problem 8:
• The statement is \(p

ightarrow q\) (if Kaylee gets at least a 95 on her final exam (\(p\)), she will get an A in Geometry (\(q\))).

• We are given \(p\) (Kaylee got 98 which is \(p\) since \(98\geq95\)).
• By the law of detachment (\((p

ightarrow q)\land p\Rightarrow q\)), the conclusion is valid.

• Problem 9:
• The statement is \(p

ightarrow q\) (if it is raining (\(p\)), then you must bring an umbrella (\(q\))).

• We are given \(q\) (you bring your umbrella).
• There is no valid logical rule (converse error: \(p

ightarrow q\) and \(q\) does not imply \(p\)) to conclude \(p\) (it is raining). The conclusion is invalid.

Answer:

1. Valid
2. Invalid
3. Valid
4. Invalid