To solve the problem, we need to analyze the given diagram. The problem states that the ironing board is parallel to the horizontal ground, and we need to find the measures of angles $\angle A$ through $\angle E$.

Step-by-Step Analysis:

1. Parallel Lines and Angles:

   • Since the ironing board is parallel to the ground, the top line (where the iron is placed) is parallel to the bottom line (the ground).
   • The legs of the ironing board form transversals across these parallel lines.

2. Angle Relationships:

   • When a transversal crosses two parallel lines, alternate interior angles are equal, and corresponding angles are equal.
   • Angles on a straight line sum up to $180^\circ$.

3. Identifying Angles:

   • Given that $\angle D = 36^\circ$, and $\angle A + \angle D = 180^\circ$ (because they are supplementary on a straight line), we can find $\angle A$.
   • Using alternate interior angles, $\angle A = \angle E$.
   • $\angle B = \angle D = 36^\circ$ (by alternate interior angles).
   • $\angle C = \angle D$ because $\angle C$ and $\angle D$ are vertically opposite angles.

Calculation:

• $\angle A = 180^\circ - \angle D = 180^\circ - 36^\circ = 144^\circ$
• $\angle E = \angle A = 144^\circ$ (corresponding angles)
• $\angle B = \angle D = 36^\circ$ (alternate interior angles)
• $\angle C = \angle D = 36^\circ$ (vertically opposite angles)

Final Angles:

• $\angle A = 144^\circ$
• $\angle B = 36^\circ$
• $\angle C = 36^\circ$
• $\angle D = 36^\circ$
• $\angle E = 144^\circ$

Would you like more details or have any questions? Here are some related questions to explore further:

1. How would the angle measures change if the ironing board were not parallel to the ground?
2. What other angle relationships can be derived from parallel lines and transversals?
3. How can you use the properties of supplementary angles to solve similar problems?
4. What happens to the angle measurements if one of the angles is doubled?
5. Can you calculate the lengths of the legs of the ironing board if given additional dimensions?

Tip: Always check for parallel lines and angle relationships like alternate interior, corresponding, and supplementary angles when dealing with problems involving transversals.