Question

what is \\(m\angle p\\)?
\\(m\angle p = \\) \\(^{\circ}\\)

Explanation:

🆕 New Concept Discovered: Angle Relationships in Triangles
How angles inside and outside triangles connect

Step 1: Find the vertical angle

The intersection of lines \(EQ\) and \(FP\) at point \(D\) creates vertical angles. Vertical angles are equal.

$$ m\angle PDQ = m\angle EDF = 92^\circ $$

Step 2: Use the Triangle Angle Sum Theorem

The sum of angles in any triangle is always \(180^\circ\). For triangle \(PDQ\):

$$ m\angle P + m\angle Q + m\angle PDQ = 180^\circ $$

Substitute the known values into the equation:

$$ m\angle P + 41^\circ + 92^\circ = 180^\circ $$

Step 3: Solve for \(m\angle P\)

Combine the known angles:

$$ m\angle P + 133^\circ = 180^\circ $$

Subtract \(133^\circ\) from both sides:

$$ m\angle P = 180^\circ - 133^\circ $$

$$ m\angle P = 47^\circ $$

Answer:

\(47\)