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The problem gives us a triangle with some angle measures and side lengths and asks for the value of \( p \). Here’s a breakdown of the problem:

We are given the following:

1. \( AB = AD = 25 \)
2. \( m \angle ACB = 6p \)
3. \( m \angle ACD = p + 50^\circ \)

The diagram shows a triangle \( ACB \) with a perpendicular from point \( D \) to side \( BC \).

### Steps:

1. Since \( AB = AD \), triangle \( ABD \) is isosceles, and \( \angle ADB = \angle ABD = 90^\circ \), as given by the right angle marks.

2. The total sum of angles in any triangle is \( 180^\circ \). Applying this to triangle \( ACD \), we get the equation for the angles in that triangle:
   \[
   6p + (p + 50^\circ) + 90^\circ = 180^\circ
   \]
   Simplifying:
   \[
   6p + p + 50^\circ + 90^\circ = 180^\circ
   \]
   \[
   7p + 140^\circ = 180^\circ
   \]
   \[
   7p = 40^\circ
   \]
   \[
   p = \frac{40^\circ}{7} \approx 5.71^\circ
   \]

Thus, \( p \approx 5.71^\circ \).

### Would you like further details on any part of this solution?

Here are 5 related questions:
1. How can we verify the angle sum property of a triangle?
2. What are the properties of isosceles triangles that help solve this?
3. How do you solve linear equations involving angles?
4. Can we solve this using trigonometric identities?
5. How does the perpendicular from \( A \) to \( BC \) affect the problem?

**Tip:** In isosceles triangles, knowing two angles or two sides gives enough information to determine the remaining angles.

If AB = AD = 25, m∠ACB = 6p, and m∠ACD = p + 50°, what is the value of p?

Learn how to solve for the value of p in a triangle where AB = AD, and the angles are related by linear equations. This problem applies key concepts of geometry, including the angle sum property and properties of isosceles triangles. Suitable for students in Grades 8-10.

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