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To solve for the measures of the angles in this diagram, we need to interpret the given information carefully.

### Given Information and Analysis
1. \( AB \parallel FD \): This tells us that \( AB \) and \( FD \) are parallel lines.
2. \( \overline{EC} \) is the perpendicular bisector of \( \overline{AD} \): This means \( \overline{EC} \) forms right angles with \( \overline{AD} \), and it divides \( \overline{AD} \) into two equal parts.
3. \( m\angle BAD = 27^\circ \): The measure of angle \( \angle BAD \) is \( 27^\circ \).
4. \( \angle 5 \cong \angle 6 \cong \angle 7 \): These angles are congruent, meaning they have equal measures.

### Step-by-Step Solution

#### 1. Determine \( m\angle BAD \) and \( m\angle 7 \)
Since \( m\angle BAD = 27^\circ \), we have this angle's measure directly. Given that \( \angle 5 \cong \angle 6 \cong \angle 7 \), \( m\angle 7 = 27^\circ \) as well.

#### 2. Find \( m\angle 1 \)
Since \( \overline{EC} \) is the perpendicular bisector of \( \overline{AD} \), we know \( \angle AEC \) and \( \angle CED \) are both \( 90^\circ \).

Because \( \angle 1 \) and \( \angle BAD \) form a linear pair along the line \( \overline{AB} \), they must add up to \( 180^\circ \):
\[
m\angle 1 = 180^\circ - m\angle BAD = 180^\circ - 27^\circ = 153^\circ
\]

#### 3. Find \( m\angle 2 \)
Since \( AB \parallel FD \), and \( \angle 1 \) and \( \angle 2 \) are corresponding angles, they are equal. Therefore:
\[
m\angle 2 = m\angle 1 = 153^\circ
\]

#### 4. Determine \( m\angle 3 \) and \( m\angle 4 \)
Using the fact that \( \overline{EC} \) is the perpendicular bisector of \( \overline{AD} \), \( \angle CED = 90^\circ \). Since angles \( \angle 3 \) and \( \angle 4 \) are adjacent to \( \angle CED \) and form a straight line, they each measure \( 90^\circ \).

#### 5. Find \( m\angle 5 \) and \( m\angle 6 \)
Since \( \angle 5 \cong \angle 6 \cong \angle 7 \), and we already know \( m\angle 7 = 27^\circ \), both \( m\angle 5 = 27^\circ \) and \( m\angle 6 = 27^\circ \).

### Summary of Angle Measures:
- \( m\angle BAD = 27^\circ \)
- \( m\angle 1 = 153^\circ \)
- \( m\angle 2 = 153^\circ \)
- \( m\angle 3 = 90^\circ \)
- \( m\angle 4 = 90^\circ \)
- \( m\angle 5 = 27^\circ \)
- \( m\angle 6 = 27^\circ \)
- \( m\angle 7 = 27^\circ \)

Would you like further details on any of these steps, or do you have questions about specific parts?

Here are five related questions for further practice:
1. How would the angle measures change if \( AB \) and \( FD \) were not parallel?
2. If \( m\angle BAD \) were a different value, how would that impact the measures of other angles?
3. What if \( \overline{EC} \) was not the perpendicular bisector of \( \overline{AD} \); how would this affect the solution?
4. How can we prove that \( \angle 1 \) and \( \angle 2 \) are corresponding angles due to parallel lines?
5. Why are \( \angle 3 \) and \( \angle 4 \) each \( 90^\circ \) based on the perpendicular bisector?

**Tip:** Always remember to use parallel line properties, such as corresponding angles, to find unknown angles when parallel lines are involved in a geometric configuration.

Determine the measures of each of the numbered angles based on the given information in the diagram.

This problem guides students through determining angle measures in a geometric figure with parallel lines and a perpendicular bisector. Using concepts like corresponding angles, congruent angles, and linear pairs, students can solve for all angles. Suitable for Grade 10 students.

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