https://file.aiquickdraw.com/mathbot/file/1727304434420r51dugu4.jpg

The image shows a triangle with its angles labeled as:

- \( p + 32^\circ \) for one angle,
- \( p + 20^\circ \) for another angle,
- \( 2p \) for the third angle.

To find the value of \( p \), we can use the **triangle angle sum theorem**, which states that the sum of the interior angles of a triangle is always \( 180^\circ \). Therefore, we can set up the equation as follows:

\[
(p + 32^\circ) + (p + 20^\circ) + (2p) = 180^\circ
\]

Now, let's simplify and solve for \( p \):

1. Combine like terms:

\[
p + p + 2p + 32^\circ + 20^\circ = 180^\circ
\]

\[
4p + 52^\circ = 180^\circ
\]

2. Subtract \( 52^\circ \) from both sides:

\[
4p = 180^\circ - 52^\circ
\]

\[
4p = 128^\circ
\]

3. Divide by 4:

\[
p = \frac{128^\circ}{4} = 32^\circ
\]

So, the value of \( p \) is \( 32^\circ \).

Would you like more details on this solution or have any questions?

### Related Questions:
1. What are the individual angle measures in the triangle?
2. How can you verify the angles add up to 180 degrees?
3. How is the triangle angle sum theorem used in geometry?
4. What happens if one angle of a triangle is 90 degrees?
5. Can a triangle have two angles that are equal?

**Tip:** Always check if the angles in a triangle add up to 180° when solving for missing angles.

What is the value of p in a triangle where the angles are labeled as p+32°, p+20°, and 2p?

In this problem, you will learn how to solve for the variable p in a triangle with angles labeled p+32°, p+20°, and 2p. By using the Triangle Angle-Sum Theorem, which states the sum of a triangle's interior angles is always 180°, we can set up an equation and solve step-by-step. This problem is suitable for students in Grades 8-10.

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