Question

the diagram shows a convex polygon. what is the value of p?

Explanation:

Step1: Find interior angles

First, calculate each interior angle by subtracting the exterior angle from $180^\circ$:
Interior angle 1: $180^\circ - 118^\circ = 62^\circ$
Interior angle 2: $180^\circ - 140^\circ = 40^\circ$
Interior angle 3: $180^\circ - 34p$

Step2: Sum interior angles of triangle

The figure is a triangle, so the sum of interior angles is $180^\circ$. Set up the equation:
$$62 + 40 + (180 - 34p) = 180$$

Step3: Simplify and solve for p

Simplify the left-hand side:
$$282 - 34p = 180$$
Rearrange to isolate the term with $p$:
$$34p = 282 - 180$$
$$34p = 102$$
Solve for $p$:
$$p = \frac{102}{34}$$

Answer:

$p = 3$