Question

complete the two-column proof using the selections given. write the letter next to the number it corresponds with. given: m is the midpoint of (overline{gh}) prove: (x = 2) (\boldsymbol{g}) (\boldsymbol{5x}) (\boldsymbol{m}) (\boldsymbol{3x + 4}) (\boldsymbol{h}) statements reasons 1. m is the midpoint of (overline{gh}) 1. 2. (overline{gm} cong overline{mh}) 2. 3. 3. definition of congruent segments 4. (5x = 3x + 4) 4. substitution property of equality 5. 5. subtraction property of equality 6. 6. division property of equality a. distributive property b. given c. (x = 2) d. definition of midpoint e. (gm = mh) f. (2x = 4)

Explanation:

Step1: Reason for Statement 1

Statement 1 is given, so the reason is "Given" (option B).

Step2: Reason for Statement 2

If M is the midpoint of \(\overline{GH}\), by the definition of midpoint, \(GM\cong MH\), so the reason is "Definition of Midpoint" (option D).

Step3: Statement for Reason 3

The definition of congruent segments states that if two segments are congruent, their lengths are equal. So the statement is \(GM = MH\) (option E).

Step4: Reason for Statement 4

We substitute \(GM = 5x\) and \(MH=3x + 4\) into \(GM = MH\) to get \(5x=3x + 4\), so the reason is "Substitution Property of Equality" (already given, but we match the statement).

Step5: Statement for Reason 5

Using the Subtraction Property of Equality, subtract \(3x\) from both sides of \(5x=3x + 4\). So \(5x-3x=3x + 4-3x\), which simplifies to \(2x = 4\) (option F).

Step6: Statement for Reason 6

Using the Division Property of Equality, divide both sides of \(2x = 4\) by 2. So \(\frac{2x}{2}=\frac{4}{2}\), which gives \(x = 2\) (option C).

Answer:

1. B
2. D
3. E
4. (already matched with substitution, but as per the table, the statement is \(5x = 3x+4\) with reason 4)
5. F
6. C

(For the two - column proof, filling the letters as follows:

1. B
2. D
3. E
4. (the statement \(5x = 3x + 4\) has reason 4)
5. F
6. C)