Paragraph Proofs

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
Answer: therefore
Therefore introduces a new claim that follows from what came before. Because, since and by definition of introduce reasons.
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
Answer: The definition of an odd integer
An odd integer is, by definition, one that can be written as $2m + 1$ for an integer $m$ . The sentence uses that definition.
3.
$\angle 1$ and $\angle 2$ form a linear pair, and $\angle 1 = 72^\circ$ . How many degrees is $\angle 2$ ?
Answer: 108
Since the angles form a linear pair, $\angle 1 + \angle 2 = 180^\circ$ , so $\angle 2 = 180^\circ - 72^\circ = 108^\circ$ .
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?
Answer: 21
With $n = 20$ the sum is $3(n + 1) = 3 \times 21 = 63$ . Check: $20 + 21 + 22 = 63$ . The number is the middle one, 21.
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 $\angle 3$ .
Answer: 58
1. $\angle 1$ and $\angle 2$ form a linear pair. How many degrees is $\angle 1 + \angle 2$ ? 180
2. So how many degrees is $\angle 2$ ? 122
3. $\angle 2$ and $\angle 3$ also form a linear pair. How many degrees is $\angle 3$ ? 58
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 $b = 2n + 1$ .
Answer: 18
1. Since $7 = 2m + 1$ , what is $m$ ? 3
2. Since $11 = 2n + 1$ , what is $n$ ? 5
3. What is $m + n + 1$ ? 9
4. So $a + b = 2(m + n + 1)$ equals what? 18
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 $a = b$ .
Answer: Adding gives $a + b = 2m + 2n + 2 = 2(m + n + 1)$ .
Only the sum $a + b = 2m + 2n + 2 = 2(m + n + 1)$ connects the start to the last sentence. $2(m + n) + 1$ is wrong algebra, $a = b$ was never given, and an example proves nothing in general.
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
Answer: Sentence 4
Sentences 1 to 3 are correct. Sentence 4 draws the wrong conclusion: 2 times an integer is even by definition, not odd.
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?
Answer: 28
$4x - 7 + 7 = 21 + 7$ , so $4x = 28$ . Dividing by 4 then gives $x = 7$ , the last row of the table.
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 $kb$ ?
Answer: 15
$6 = 2 \times 3$ , so $k = 3$ and $kb = 3 \times 5 = 15$ . Check: $ab = 30 = 2 \times 15$ .
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.
Answer: 6
1. Row 2 (distributive property): $2x + \square = 18$ . What number goes in the box? 6
2. Row 3 (subtraction property of equality): $2x = \square$ 12
3. Row 4 (division property of equality): $x = \square$ 6
4. How many rows does the two-column proof have, counting the given? 4
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.
Answer: 31, 32, 33
1. Since $3(n + 1) = 96$ , what is $n + 1$ , the middle integer? 32
2. What is the smallest integer $n$ ? 31
3. What is the largest integer $n + 2$ ? 33
13.
Follow the paragraph proof that an even integer times any integer is even with $a = 14$ and $b = 9$ : write $a = 2k$ , then $ab = 2(kb)$ .
Answer: 126
1. Since $14 = 2k$ , what is $k$ ? 7
2. What is $kb$ ? 63
3. So $ab = 2(kb)$ equals what? 126
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 $a = b$
Answer: Using the same $k$ for both numbers proves it only when $a = b$
Writing both numbers with the same $k$ forces $a = b$ , so the proof only covers sums like $7 + 7$ . The algebra itself is correct, and $4k + 2$ is even. The fix is $a = 2m + 1$ , $b = 2n + 1$ .
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 $21 + 22 + 23 + 24 + 25$ .
Answer: 115
With $n = 21$ : $5(n + 2) = 5 \times 23 = 115$ . Check: $21 + 22 + 23 + 24 + 25 = 115$ .
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.
Answer: 20
1. Subtract $3x$ from both sides and add 30 to both sides. The equation becomes $\square = 2x$ . What number is in the box? 40
2. What is $x$ ? 20
3. How many degrees is each vertical angle? 70
4. The angle next to it forms a linear pair with it. How many degrees is that angle? 110
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 $\angle 3$ .
Answer: 55
1. Since the angles form a linear pair, $6x + 60 = 180$ . What is $6x$ ? 120
2. What is $x$ ? 20
3. How many degrees is $\angle 1$ ? 55
4. How many degrees is $\angle 3$ ? 55
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.
Answer: 60
1. Add: $2n + (2n + 2) + (2n + 4) = 6n + \square$ . What number is in the box? 6
2. So the sum is $6(n + 1)$ . For 18, 20, 22 we have $2n = 18$ . What is $n$ ? 9
3. What is $6(n + 1)$ ? 60
4. Divide the sum by 6. Which whole number do you get? 10