Let's walk through the questions on your geometry worksheet and solve them step by step.

1. Points $P$, $Q$, and $R$ are collinear.

a. Find $PQ$:
From the number line, $P = -2$ and $Q = 0$. The distance between $P$ and $Q$ is:
$PQ = |0 - (-2)| = 2$

b. If $R$'s coordinate is 7, why is $PQ$ not congruent to $QR$?
We found that $PQ = 2$. For $QR$, the distance is:
$QR = |7 - 0| = 7$
Since $PQ = 2$ and $QR = 7$, they are not congruent because their lengths are different.

c. What must the coordinate of $R$ be in order for $Q$ to be the midpoint of $PR$?
If $Q$ is the midpoint of $PR$, the distance from $P$ to $Q$ must be equal to the distance from $Q$ to $R$.
Thus,
$PQ = QR$
Since $PQ = 2$, we need $QR = 2$ as well. Therefore,
$R = Q + QR = 0 + 2 = 2$
So, the coordinate of $R$ must be 2.

2. Points $W$, $X$, and $Y$ are collinear. $WY = 25$ and the ratio of $WX$ to $XY$ is 2:3. Find $WX$.

Let $WX = 2k$ and $XY = 3k$, where $k$ is a constant.
Thus,
$WX + XY = WY$
Substitute the values:
$2k + 3k = 25$
$5k = 25$
$k = 5$
So,
$WX = 2k = 2(5) = 10$

3. The lengths of two segments are in the ratio 5:3, and the longer segment exceeds the shorter segment by 14. Find the length of the longer segment.

Let the length of the shorter segment be $3x$, and the length of the longer segment be $5x$.
The difference between the longer and shorter segment is 14:
$5x - 3x = 14$
$2x = 14$
$x = 7$
So, the length of the longer segment is:
$5x = 5(7) = 35$

4. If $CD = 5x - 7$, find the value of $x$, $CE$, and $CD$.

From the image, $CE = 2x - 3$ and $ED = 3x - 4$, and it is implied that $CD = CE + ED$.
Thus,
$CD = CE + ED$
Substitute the expressions:
$5x - 7 = (2x - 3) + (3x - 4)$
Simplify:
$5x - 7 = 5x - 7$
This is true for all $x$, so no specific value of $x$ is given. The values of $CE$ and $CD$ are left in terms of $x$.

5. If $C$ is the midpoint of $KN$, what is $KC$?

If $C$ is the midpoint, then $KC = CN$.
From the diagram, $KC = 2x + 10$ and $CN = 4x + 1$.
Since $KC = CN$:
$2x + 10 = 4x + 1$
Solve for $x$:
$10 - 1 = 4x - 2x$
$9 = 2x$
$x = \frac{9}{2} = 4.5$
Substitute $x$ back to find $KC$:
$KC = 2(4.5) + 10 = 9 + 10 = 19$

Let me know if you'd like further explanations on any of these problems or need help with the remaining ones!