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.