C. Find the measure of the sides and angies the theorem used. 1. In isosceles trapezoid MARE where 2. If $$ M E=18 $$, what is $$ A R $$ ? 3. If $$ S $$ and $$ T $$ are the midpoints of $$ M E $$ an 4. If $$ M A=25 $$ and $$ E R=39 $$, what is $$ S T $$ ? 5. If $$ S T=23 $$ and $$ M A=17 $$, what is $$ E R $$ ? 6. If $$ S T=40 $$ and $$ E R=50 $$, what is $$ M A $$ ? 7. If $$ M R=27 $$, what is $$ A E $$ ? 8. In trapezoid $$ R E A P $$ if $$ \angle R $$ and $$ \angle E $$ are 8. $$ \angle P $$ ?

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To solve the problem regarding the isosceles trapezoid MARE and trapezoid REAP, we will follow the steps outlined in the problem statement.

### Known Conditions:
1. MARE is an isosceles trapezoid.
2. $$ ME = 18 $$
3. $$ MA = 25 $$
4. $$ ER = 39 $$
5. $$ MR = 27 $$
6. $$ S $$ and $$ T $$ are midpoints of $$ ME $$ and $$ AR $$ respectively.
7. $$ ST $$ is the segment connecting midpoints $$ S $$ and $$ T $$.

### Step-by-Step Solutions:

1. **Finding $$ AR $$**:
   - In an isosceles trapezoid, the lengths of the bases are equal. Therefore, $$ AR = ME = 18 $$.

2. **Finding $$ ST $$**:
   - The length of segment $$ ST $$ in an isosceles trapezoid can be calculated using the formula:
     $$
     ST = \frac{1}{2} (ME + AR)
     $$
   - Substituting the known values:
     $$
     ST = \frac{1}{2} (18 + 18) = \frac{1}{2} \times 36 = 18
     $$

3. **Finding $$ ER $$**:
   - Given $$ ST = 23 $$ and $$ MA = 17 $$, we can use the relationship:
     $$
     ST = \frac{1}{2} (ME + AR)
     $$
   - Rearranging gives:
     $$
     ME + AR = 2 \times ST
     $$
   - Substituting $$ ST = 23 $$:
     $$
     ME + AR = 2 \times 23 = 46
     $$
   - Since $$ ME = 18 $$, we can find $$ AR $$:
     $$
     18 + AR = 46 \implies AR = 46 - 18 = 28
     $$

4. **Finding $$ MA $$**:
   - Given $$ ST = 40 $$ and $$ ER = 50 $$:
     $$
     ST = \frac{1}{2} (ME + AR)
     $$
   - Rearranging gives:
     $$
     ME + AR = 2 \times ST
     $$
   - Substituting $$ ST = 40 $$:
     $$
     ME + AR = 2 \times 40 = 80
     $$
   - Since $$ ER = 50 $$, we can find $$ MA $$:
     $$
     MA + ER = 80 \implies MA + 50 = 80 \implies MA = 80 - 50 = 30
     $$

5. **Finding $$ AE $$**:
   - Given $$ MR = 27 $$:
     - In an isosceles trapezoid, the legs are equal, so $$ MA = AE $$.
     - Therefore, $$ AE = 27 $$.

6. **Finding angles in trapezoid $$ REAP $$**:
   - In trapezoid $$ REAP $$, if $$ \angle R $$ and $$ \angle E $$ are equal (as it is an isosceles trapezoid), then $$ \angle P $$ can be calculated using the property that the sum of angles in a quadrilateral is $$ 360^\circ $$:
     $$
     \angle R + \angle E + \angle A + \angle P = 360^\circ
     $$
   - Since $$ \angle R = \angle E $$, let $$ \angle R = \angle E = x $$:
     $$
     2x + \angle A + \angle P = 360^\circ
     $$
   - If we assume $$ \angle A $$ is also equal to $$ x $$ (for simplicity), we can solve for $$ \angle P $$:
     $$
     3x + \angle P = 360^\circ \implies \angle P = 360^\circ - 3x
     $$

### Summary of Results:
1. $$ AR = 18 $$
2. $$ ST = 18 $$
3. $$ ER = 28 $$
4. $$ MA = 30 $$
5. $$ AE = 27 $$
6. $$ \angle P = 360^\circ - 3x $$ (where $$ x $$ is the measure of angles $$ R $$ and $$ E $$)

This concludes the solution to the problem. If you have any further questions or need additional calculations, feel free to ask!
Here are the answers to the questions: 1. $$ AR = 18 $$ 2. $$ ST = 18 $$ 3. $$ ER = 28 $$ 4. $$ MA = 30 $$ 5. $$ AE = 27 $$ 6. $$ \angle P = 360^\circ - 3x $$ (where $$ x $$ is the measure of angles $$ R $$ and $$ E $$)

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