Question

find the longest side of $\triangle hij$.

Explanation:

Step1: Sum triangle angles to 180°

$$(p+72^\circ) + 23p + 30p = 180^\circ$$

Step2: Simplify to solve for $p$

$$54p + 72^\circ = 180^\circ$$
$$54p = 180^\circ - 72^\circ = 108^\circ$$
$$p = \frac{108^\circ}{54} = 2^\circ$$

Step3: Calculate each angle

$\angle H = 2^\circ + 72^\circ = 74^\circ$
$\angle J = 23 \times 2^\circ = 46^\circ$
$\angle I = 30 \times 2^\circ = 60^\circ$

Step4: Match largest angle to side

Largest angle is $\angle H$, opposite side $IJ$.

Answer:

The longest side is $\overline{IJ}$