Paragraph Proofs

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Show your work for each problem.

1.
In a paragraph proof, which signal word introduces a new claim (a conclusion) rather than a reason?
1. a)since
2. b)therefore
3. c)because
4. d)by definition of
2.
A paragraph proof contains the sentence: *Since $a$ is odd, $a = 2m + 1$ for some integer $m$ .* Which reason does this sentence use?
1. a)The definition of an odd integer
2. b)The subtraction property of equality
3. c)The conclusion of the proof
4. d)The vertical angles theorem
3.
$\angle 1$ and $\angle 2$ form a linear pair, and $\angle 1 = 72^\circ$ . How many degrees is $\angle 2$ ?
4.
A paragraph proof shows that three consecutive integers $n$ , $n + 1$ , $n + 2$ add up to $3(n + 1)$ . For $20 + 21 + 22$ , the sum equals 3 times which number?
5.
Two lines intersect. $\angle 1 = 58^\circ$ , $\angle 2$ is next to it and $\angle 3$ is opposite $\angle 1$ . Follow the paragraph proof of the vertical angles theorem to find $\angle 3$ .
6.
Follow the paragraph proof that odd + odd is even with the odd numbers $a = 7$ and $b = 11$ , written as $a = 2m + 1$ and $b = 2n + 1$ .
7.
One sentence is missing from this paragraph proof that the sum of two odd integers is even: *Let $a$ and $b$ be odd. Then $a = 2m + 1$ and $b = 2n + 1$ for integers $m$ and $n$ . [ ? ] Since $m + n + 1$ is an integer, $a + b$ is even.* Which sentence fills the gap?
1. a)So $a + b = 2(m + n) + 1$ .
2. b)For example, $3 + 5 = 8$ is even.
3. c)Adding gives $a + b = 2m + 2n + 2 = 2(m + n + 1)$ .
4. d)Therefore $a = b$ .
8.
Which sentence of this paragraph proof is wrong? (1) Let $a$ be even and $b$ any integer, so $a = 2k$ for an integer $k$ . (2) Then $ab = 2kb$ . (3) So $ab = 2(kb)$ . (4) Since $kb$ is an integer, $ab$ is odd.
1. a)Sentence 2
2. b)Sentence 1
3. c)Sentence 3
4. d)Sentence 4
9.
A two-column proof starts with the given $4x - 7 = 21$ and ends with $x = 7$ . In the paragraph version the second sentence reads: *Adding 7 to both sides gives $4x = \square$ by the addition property of equality.* What number goes in the box?
10.
The paragraph proof that an even integer times any integer is even writes $a = 2k$ and gets $ab = 2(kb)$ . For $a = 6$ and $b = 5$ , what is the value of $kb$ ?
11.
Turn this paragraph proof into a two-column proof: *We are given $2(x + 3) = 18$ . By the distributive property, $2x + 6 = 18$ . Subtracting 6 from both sides gives $2x = 12$ . Dividing both sides by 2 gives $x = 6$ .* Fill in the statements.
12.
Three consecutive integers add up to 96. Use the result of the paragraph proof, $n + (n + 1) + (n + 2) = 3(n + 1)$ , to find them.
13.
Follow the paragraph proof that an even integer times any integer is even with $a = 14$ and $b = 9$ : write $a = 2k$ , then $ab = 2(kb)$ .
14.
A student writes: *Let $a$ and $b$ be odd, so $a = 2k + 1$ and $b = 2k + 1$ . Then $a + b = 4k + 2 = 2(2k + 1)$ , so $a + b$ is even.* What is the gap in this proof?
1. a)A proof must use numbers, not letters
2. b)There is no gap, the proof is complete
3. c) $4k + 2$ is always odd
4. d)Using the same $k$ for both numbers proves it only when $a = b$
15.
A paragraph proof shows that five consecutive integers add up to $n + (n + 1) + (n + 2) + (n + 3) + (n + 4) = 5(n + 2)$ . Use this result to find $21 + 22 + 23 + 24 + 25$ .
16.
Two vertical angles measure $(3x + 10)^\circ$ and $(5x - 30)^\circ$ . Reason as in a paragraph proof: since vertical angles are equal, $3x + 10 = 5x - 30$ . Find $x$ , the angle, and the angle next to it.
17.
Two lines intersect. $\angle 1 = (2x + 15)^\circ$ and $\angle 2 = (4x + 45)^\circ$ form a linear pair, and $\angle 3$ is vertical to $\angle 1$ . Find $x$ , $\angle 1$ and $\angle 3$ .
18.
Prove that the sum of three consecutive even integers $2n$ , $2n + 2$ , $2n + 4$ is divisible by 6, then test the result with 18, 20 and 22.