6.	Let ABCD be a parrallelgram in which the bisector of angle ABC intercects AD at P given PD = 65 and BP = 78 = PC. Find AB

Given that \(ABCD\) is a parallelogram, and the angle bisector of \(\angle ABC\) intersects \(AD\) at \(P\), with \(PD = 65\), \(BP = 78\), and \(PC = 78\), we are to find the length of \(AB\).

### Step-by-Step Solution:

1. **Properties of the Parallelogram:**
   Since \(ABCD\) is a parallelogram, we have:
   - \(AB = CD\)
   - \(AD = BC\)

2. **Angle Bisector Theorem Application:**
   The angle bisector of \(\angle ABC\) divides the opposite side \(DC\) in the ratio of the adjacent sides \(AB\) and \(BC\). However, in this case, it bisects the parallelogram's diagonal \(AC\) at point \(P\).

   Since \(BP = PC = 78\), \(P\) is the midpoint of diagonal \(AC\).

3. **Using Triangle Properties:**
   Let's focus on \(\triangle ABP\) and \(\triangle DPC\).

   Since \(P\) is the midpoint of \(AC\), and \(BP = PC = 78\), the diagonal \(AC\) is \(AC = BP + PC = 78 + 78 = 156\).

   Also, in parallelogram \(ABCD\), diagonals bisect each other. Thus, the diagonals \(AC\) and \(BD\) bisect each other at point \(P\). Since \(P\) is the midpoint of both diagonals, \(AP = PC = 78\) and \(PD = PB = 65\).

4. **Calculating the Length of AB:**

   Consider \(\triangle ABP\), where \(AB = x\), \(BP = 78\), and \(PD = 65\).

   Using the fact that diagonals bisect each other in a parallelogram:
   \[
   \frac{AB}{PD} = \frac{BP}{PC}
   \]

   Substituting the values:
   \[
   \frac{x}{65} = \frac{78}{78} \implies \frac{x}{65} = 1 \implies x = 65
   \]

Therefore, \(AB = 65\).

### Final Answer:

\[
\boxed{AB = 65}
\]

Would you like to go over any specific part of this solution, or do you have any other questions?

### Related Questions:
1. What are the properties of diagonals in a parallelogram?
2. How does the angle bisector theorem apply to a triangle?
3. How do you determine the length of a side in a parallelogram?
4. Can you find the area of parallelogram \(ABCD\) if given more information?
5. What are the conditions for a quadrilateral to be a parallelogram?

**Tip:** Remember that in a parallelogram, opposite sides are equal, and diagonals bisect each other. This can be very useful in solving geometry problems involving parallelograms.

Let ABCD be a parallelogram in which the bisector of angle ABC intersects AD at P given PD = 65 and BP = 78 = PC. Find AB

Learn how to find the length of side AB in a parallelogram ABCD, where the angle bisector of angle ABC intersects AD at P, with PD = 65 and BP = PC = 78. This problem covers key geometry concepts such as the Angle Bisector Theorem and properties of parallelograms. Suitable for students in Grades 9-11.

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