in the diagram below, $overline{ac}congoverline{ce}$ and $d$ is the mid - point of $overline{ce}$. if $ce = 10x + 18$, $de=7x - 1$, and $bc = 9x - 2$, find $ab$.
13. classify $angle1$ and $angle2$ using all relationships that apply.
14. classify $angle1$ and $angle2$ using all relationships that apply.
15. if $mangle tuv=(9x + 1)^{circ}$, $mangle tuw=(7x - 9)^{circ}$, and $mangle wuv=(5x - 11)^{circ}$, find the value of $x$.
16. find the value of $x$.

Question

Accepted Answer

### Explanation:
#### Step1: Find the value of $x$ from the mid - point information
Since $D$ is the mid - point of $CE$, then $CE = 2DE$. Given $CE=10x + 18$ and $DE = 7x-1$, we have the equation $10x + 18=2(7x - 1)$.
$$
\begin{align*}
10x+18&=14x-2\
18 + 2&=14x-10x\
20&=4x\
x&=5
\end{align*}
$$
#### Step2: Find the lengths of $AC$ and $BC$
First, find $AC$. Since $AC = CE$ and $CE=10x + 18$, substituting $x = 5$, we get $CE=10\times5+18=50 + 18=68$, so $AC = 68$.
Second, find $BC$. Given $BC=9x - 2$, substituting $x = 5$, we get $BC=9\times5-2=45-2 = 43$.
#### Step3: Find the length of $AB$
Since $AC=AB + BC$, then $AB=AC - BC$. Substituting $AC = 68$ and $BC = 43$, we get $AB=68-43 = 25$.

### Answer:
$25$

### Explanation for question 13:
Adjacent angles share a common side and a common vertex. Vertical angles are opposite angles formed by the intersection of two lines. Complementary angles add up to $90^{\circ}$, supplementary angles add up to $180^{\circ}$, and a linear pair is a pair of adjacent supplementary angles. For $\angle1$ and $\angle2$ in question 13, they are vertical angles as they are opposite angles formed by the intersection of two lines.
### Answer for question 13:
Vertical

### Explanation for question 14:
$\angle1$ and $\angle2$ share a common vertex and a common side, so they are adjacent. They are also complementary because the angle formed by the two intersecting lines is a right - angle, and $\angle1+\angle2 = 90^{\circ}$.
### Answer for question 14:
Adjacent, Complementary

### Explanation for question 15:
Since $m\angle TUV=m\angle TUW + m\angle WUV$, we have the equation $(9x + 1)=(7x-9)+(5x - 11)$.
$$
\begin{align*}
9x+1&=7x-9 + 5x-11\
9x+1&=12x-20\
20 + 1&=12x-9x\
21&=3x\
x&=7
\end{align*}
$$
### Answer for question 15:
$7$

### Explanation for question 16:
The two angles $(7x - 1)^{\circ}$ and $(6x - 1)^{\circ}$ are vertical angles, and vertical angles are equal. So we set up the equation $7x-1=6x - 1$.
$$
\begin{align*}
7x-6x&=-1 + 1\
x&=0
\end{align*}
$$
### Answer for question 16:
$0$

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QUESTION IMAGE

Question

Explanation:

Step1: Find the value of \(x\) from the mid - point information

Step2: Find the lengths of \(AC\) and \(BC\)

Step3: Find the length of \(AB\)

Answer: