Question

fill in the missing statements for the following two - column proof. given: c is the midpoint of $overline{ad}$. prove: $x = 9$. statements reasons 1) c is the midpoint of $overline{ad}$. 1) a. ? 2) $overline{ac}congoverline{cd}$ 2) b. ? 3) $ac = cd$ 3) $cong$ segments have equal length. 4) $5x=3x + 18$ 4) c. ? 5) ? 5) d. subtraction property of equality 6) $x = 9$ 6) e. ? statements reasons 1) c is the midpoint of $overline{ad}$. 1) a. definition of midpoint segment addition postulate transitive property of congruence given

Explanation:

Step1: Given information

The first statement "C is the midpoint of $\overline{AD}$" is given in the problem - statement, so the reason for this statement is "Given".

Step2: Definition of mid - point

If a point C is the mid - point of a line segment $\overline{AD}$, then by the definition of a mid - point, the two sub - segments $\overline{AC}$ and $\overline{CD}$ are congruent. So the reason for $\overline{AC}\cong\overline{CD}$ is "Definition of midpoint".

Step3: Congruent segments have equal length

Since $\overline{AC}\cong\overline{CD}$, we can say $AC = CD$ because congruent segments have equal length.

Step4: Substitute segment lengths

We are given that $AC = 5x$ and $CD=3x + 18$, so by substitution, $5x=3x + 18$.

Step5: Apply subtraction property of equality

Subtract $3x$ from both sides of the equation $5x=3x + 18$. We get $5x-3x=3x + 18-3x$, which simplifies to $2x=18$.

Step6: Apply division property of equality

Divide both sides of the equation $2x = 18$ by 2. So $x=\frac{18}{2}=9$.

Answer:

a. Given
b. Definition of midpoint
c. Substitution
d. $2x = 18$
e. Division property of equality