Question

find the shortest side of $\triangle pqr$.

Explanation:

Step1: Sum angles to 180°

$$(w+32^\circ)+(w+23^\circ)+(w+29^\circ)=180^\circ$$

Step2: Simplify to solve for w

$$3w + 84^\circ=180^\circ$$
$$3w=180^\circ-84^\circ=96^\circ$$
$$w=\frac{96^\circ}{3}=32^\circ$$

Step3: Calculate each angle

$\angle Q = 32^\circ+32^\circ=64^\circ$
$\angle R = 32^\circ+23^\circ=55^\circ$
$\angle P = 32^\circ+29^\circ=61^\circ$

Step4: Match smallest angle to side

The smallest angle is $\angle R=55^\circ$, which is opposite side $PQ$.

Answer:

The shortest side of $\Delta PQR$ is $PQ$.