The image shows a problem about a regular pentagon $ABCDE$ and asks to calculate the missing angles marked as $a$, $d$, $e$, and $f$. Let's analyze and solve it step by step.

Step 1: Properties of a regular pentagon

In a regular pentagon:

• All sides are equal.
• Each interior angle is given by the formula: $\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}$ For a pentagon ($n = 5$): $\text{Interior Angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ$ So, each interior angle in the pentagon is $108^\circ$.

Step 2: Identify the angles

1. Angle $a$ is at the center of the pentagon. The angle at the center for a regular polygon can be calculated as: $\text{Central Angle} = \frac{360^\circ}{n} = \frac{360^\circ}{5} = 72^\circ$ Thus, $a = 72^\circ$.

2. Angle $e$ forms a straight line with one of the interior angles at vertex $C$. Since $\angle C = 108^\circ$ and the sum of angles on a straight line is $180^\circ$: $e = 180^\circ - 108^\circ = 72^\circ$

3. Angle $f$ forms a linear pair with $e$. Given that $e = 72^\circ$, and knowing the total for a straight line is $180^\circ$: $f = 180^\circ - 72^\circ = 108^\circ$

4. Angle $d$ is vertically opposite to angle $e$, so $d = e = 72^\circ$.

Final Answer:

• $a = 72^\circ$
• $d = 72^\circ$
• $e = 72^\circ$
• $f = 108^\circ$

Would you like further clarification or details on any step?

Here are 5 related questions to extend your understanding:

1. How would you calculate the interior angle for any regular polygon?
2. How do you find the sum of exterior angles for any polygon?
3. Can you prove that angles opposite each other (like $d$ and $e$) are equal in this figure?
4. What is the relationship between the exterior angle and the central angle in a regular polygon?
5. How would you solve for the missing angles if the pentagon were irregular?

Tip: Always check for symmetry in regular polygons—it often simplifies angle calculations!