Two-Column Proofs

Show your work for each problem.

1.
In a two-column proof, which reason is written next to statement 1?
1. a)Reflexive Property
2. b)Definition of supplementary angles
3. c)Substitution
4. d)Given
Answer: Given
A proof starts from the information we are told, so statement 1 is justified by "Given".
2.
Given: $3x + 5 = 20$ . Statement 2 uses the Subtraction Property of Equality (subtract 5 from both sides): $3x = \square$ . What number goes in the box?
Answer: 15
$3x + 5 - 5 = 20 - 5$ gives $3x = 15$ , justified by the Subtraction Property of Equality.
3.
Given: $x + 9 = 14$ . Prove: $x = 5$ . Complete the two-column proof.
Answer: 5
1. Statement 1 is $x + 9 = 14$ (Given). Which number do you subtract from both sides? 9
2. Statement 2 (Subtraction Property of Equality): $x = ?$ 5
3. How many statements does the finished proof have? 2
4.
Which reason justifies the statement $AB = AB$ ?
1. a)Transitive Property
2. b)Substitution
3. c)Reflexive Property
4. d)Symmetric Property
Answer: Reflexive Property
$a = a$ is the Reflexive Property. Symmetric would turn $a = b$ into $b = a$ , Transitive needs two equations.
5.
Given: $\angle A$ and $\angle B$ are complementary, $m\angle A = 34^\circ$ . By the definition of complementary angles and the Subtraction Property of Equality, what is $m\angle B$ ? Write the number of degrees only.
Answer: 56
$m\angle A + m\angle B = 90^\circ$ , so $34^\circ + m\angle B = 90^\circ$ and $m\angle B = 56^\circ$ .
6.
Given: $\angle 1$ and $\angle 2$ are supplementary, $m\angle 1 = 120^\circ$ . Prove: $m\angle 2 = 60^\circ$ .
Answer: 60
1. By the definition of supplementary angles, $m\angle 1 + m\angle 2$ equals how many degrees? 180
2. By substitution, $120 + m\angle 2 = 180$ . After the Subtraction Property of Equality, $m\angle 2 = ?$ 60
7.
Statement 1: $\frac{x}{4} = 3$ (Given). Statement 2: $x = 12$ . Which reason justifies statement 2?
1. a)Distributive Property
2. b)Substitution
3. c)Multiplication Property of Equality
4. d)Division Property of Equality
Answer: Multiplication Property of Equality
Both sides were multiplied by 4: $\frac{x}{4} \cdot 4 = 3 \cdot 4$ . That is the Multiplication Property of Equality, even though it undoes a division.
8.
Given: $4(x + 3) = 36$ . Prove: $x = 6$ .
Answer: 6
1. Distributive Property: statement 2 is $4x + \square = 36$ . What number? 12
2. Subtraction Property of Equality: $4x = ?$ 24
3. Division Property of Equality: $x = ?$ 6
9.
Points $A$ , $B$ , $C$ lie on a line in this order. $AB = 2x + 1$ , $BC = 7$ and $AC = 20$ . Use the Segment Addition Postulate and prove the value of $x$ . What is $x$ ?
Answer: 6
$AB + BC = AC$ (Segment Addition Postulate), so $2x + 1 + 7 = 20$ (Substitution), $2x + 8 = 20$ , $2x = 12$ (Subtraction Property), $x = 6$ (Division Property).
10.
$\angle 1$ and $\angle 3$ are vertical angles with $m\angle 1 = (4x + 5)^\circ$ and $m\angle 3 = 65^\circ$ . Prove: $x = 15$ .
Answer: 15
1. Vertical Angles Theorem: $4x + 5 = \square$ . What number? 65
2. Subtraction Property of Equality: $4x = ?$ 60
3. Division Property of Equality: $x = ?$ 15
11.
Proof: 1. $5x - 2 = 3x + 8$ (Given); 2. $2x - 2 = 8$ (Subtraction Property of Equality); 3. $2x = 10$ (?); 4. $x = 5$ (Division Property of Equality). Which reason justifies step 3?
1. a)Division Property of Equality
2. b)Addition Property of Equality
3. c)Subtraction Property of Equality
4. d)Transitive Property
Answer: Addition Property of Equality
From $2x - 2 = 8$ to $2x = 10$ , 2 was added to both sides: the Addition Property of Equality.
12.
Given: $\frac{x}{2} - 3 = 4$ . Prove the value of $x$ using the Addition Property and then the Multiplication Property of Equality. What is $x$ ?
Answer: 14
Addition Property: $\frac{x}{2} = 7$ . Multiplication Property (multiply by 2): $x = 14$ . Check: $\frac{14}{2} - 3 = 4$ .
13.
Given: $7x - 4 = 3x + 16$ . Prove: $x = 5$ .
Answer: 5
1. Subtraction Property of Equality (subtract $3x$ ): statement 2 is $\square x - 4 = 16$ . What number? 4
2. Addition Property of Equality: $4x = ?$ 20
3. Division Property of Equality: $x = ?$ 5
14.
Find the error. 1. $\frac{x}{3} + 2 = 6$ (Given); 2. $\frac{x}{3} = 4$ (Subtraction Property of Equality); 3. $x = 12$ (Division Property of Equality). Which part of the proof is wrong?
1. a)Reason 2
2. b)Statement 2
3. c)Reason 3
4. d)Statement 3
Answer: Reason 3
All statements are true: $\frac{x}{3} = 4$ and $x = 12$ . But step 3 multiplies both sides by 3 ( $\frac{x}{3} \cdot 3 = 4 \cdot 3$ ), so reason 3 must be the Multiplication Property of Equality.
15.
$\angle 1$ and $\angle 3$ are vertical angles with $m\angle 1 = (5x - 10)^\circ$ and $m\angle 3 = (3x + 20)^\circ$ . Prove the value of $x$ , then find $m\angle 1$ . Write the number of degrees only.
Answer: 65
Vertical Angles Theorem: $5x - 10 = 3x + 20$ . Subtraction Property: $2x - 10 = 20$ . Addition Property: $2x = 30$ . Division Property: $x = 15$ . Substitution: $m\angle 1 = 5 \cdot 15 - 10 = 65^\circ$ (and $m\angle 3 = 3 \cdot 15 + 20 = 65^\circ$ ).
16.
$\angle 1$ and $\angle 2$ form a linear pair, $m\angle 1 = (3x + 10)^\circ$ and $m\angle 2 = (2x + 20)^\circ$ . Prove: $m\angle 1 = 100^\circ$ .
Answer: 100
1. A linear pair is supplementary: $(3x + 10) + (2x + 20) = ?$ 180
2. Simplify: $5x + 30 = 180$ . Subtraction Property of Equality: $5x = ?$ 150
3. Division Property of Equality: $x = ?$ 30
4. Substitution: $m\angle 1 = 3 \cdot 30 + 10 = ?$ 100
17.
Points $A$ , $B$ , $C$ lie on a line in this order. $AB = x + 4$ , $BC = 2x + 5$ and $AC = 30$ . Prove: $AB = 11$ .
Answer: 11
1. Segment Addition Postulate and Substitution: $(x + 4) + (2x + 5) = 30$ . Simplified: $3x + \square = 30$ . What number? 9
2. Subtraction Property of Equality: $3x = ?$ 21
3. Division Property of Equality: $x = ?$ 7
4. Substitution: $AB = x + 4 = ?$ 11
18.
Point $C$ lies inside the right angle $\angle ABD$ . $m\angle ABC = (2x)^\circ$ and $m\angle CBD = (3x + 15)^\circ$ . Prove: $m\angle CBD = 60^\circ$ .
Answer: 60
1. Angle Addition Postulate and the definition of a right angle: $2x + (3x + 15) = ?$ 90
2. Simplify to $5x + 15 = 90$ . Subtraction Property of Equality: $5x = ?$ 75
3. Division Property of Equality: $x = ?$ 15
4. Substitution: $m\angle CBD = 3 \cdot 15 + 15 = ?$ 60