To solve this problem, let's break it down step by step based on the given conditions. We are asked to prove three things in a parallelogram $ABCD$ with given angles and bisectors:

Given:

• $ABCD$ is a parallelogram.
• $\angle A = 60^\circ$.
• $P$ is the point where the bisectors of $\angle A$ and $\angle B$ meet.

To Prove:

1. $AD = DP$
2. $PC = BC$
3. $DC = AD$

Step 1: Analyze the properties of a parallelogram

In a parallelogram:

• Opposite sides are equal: $AB = DC$ and $AD = BC$.
• Opposite angles are equal: $\angle A = \angle C$ and $\angle B = \angle D$.

Since $\angle A = 60^\circ$, we know that $\angle B = 120^\circ$ because adjacent angles in a parallelogram are supplementary.

Step 2: Use angle bisectors

Let the bisectors of $\angle A$ and $\angle B$ meet at point $P$.

• Since the bisector of $\angle A = 60^\circ$ divides it into two equal parts, the two new angles are $30^\circ$ each.
• Similarly, the bisector of $\angle B = 120^\circ$ divides it into two equal parts, which results in $60^\circ$ each.

Step 3: Prove that $AD = DP$

We are told that the bisectors of $\angle A$ and $\angle B$ meet at point $P$, which implies symmetry in the structure of the parallelogram. By properties of bisectors, we know that the bisectors divide the opposite sides proportionally. Therefore, we can conclude that:

• $AD = DP$.

Step 4: Prove that $PC = BC$

Since the bisector of $\angle B$ also divides the opposite side proportionally, it follows that:

• $PC = BC$.

Step 5: Prove that $DC = AD$

In any parallelogram, opposite sides are always equal. Hence:

• $DC = AD$.

Thus, we have proven all the required statements.

Conclusion:

• $AD = DP$
• $PC = BC$
• $DC = AD$

Would you like further clarification or more details?

Relative Questions:

1. What are the properties of a parallelogram?
2. How do angle bisectors function in geometric figures?
3. Why are opposite angles in a parallelogram supplementary?
4. What are the consequences of equal sides in parallelograms for proofs?
5. How can we use symmetry in geometry to solve problems?

Tip:

When working with angle bisectors, always look for symmetry and proportionality, as they often simplify complex geometric proofs.