Problem 1
Answer: x = 9
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Explanation:
For now focus on triangle DEF. We need to find the missing angle D.
D+E+F = 180
D+71+42 = 180
D+113 = 180
D = 180-113
D = 67
In order for DE to be parallel to GH, the alternate interior angles EDF and GHF must be congruent
Let's see what happens when we force them to be equal
angle EDF = angle GHF
67 = 5x+22
67-22 = 5x
45 = 5x
5x = 45
x = 45/5
x = 9
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Problem 2
Answers:
Converse: "if it is winter, then it is snowing"
Inverse: "If it is not snowing, then it is not winter"
Contrapositive: "If it is not winter, then it is not snowing"
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Explanation:
The original conditional is in the form "If P, then Q". The P and Q are placeholders for logical statements.
We then have the following forms
- converse = "if Q, then P" (swap P and Q)
- inverse = "If not P, then not Q" (negate each)
- contrapositive = "If not Q, then not P" (do both inverse and converse operations)
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Problem 3
Answer: choice B) "If the sum of two angles is not 180 degrees, then they aren't supplementary"
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Explanation:
Refer to problem 2 for the general form of the inverse. We negate each of the P and Q to get our final answer.
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Problem 4
Answer: Choice C) "If I do not need an umbrella, then it is not raining"
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Explanation:
Refer to problem 2 for the general form of the contrapositive. We swap P and Q, and negate each.
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Problem 5
Answer: Choice A) "If two lines have opposite slopes, then they are perpendicular"
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Explanation:
Refer to problem 2 for the general form of the converse. We swap the positions of P and Q.
