Question

list the side lengths of $\triangle ijk$ in order from shortest to longest.

Explanation:

Step1: Sum angles to 180°

$$72 + (x+12) + (x+30) = 180$$

Step2: Simplify and solve for x

$$2x + 114 = 180 \implies 2x=66 \implies x=33$$

Step3: Calculate each angle

• $\angle K = 33+12=45^\circ$
• $\angle J = 33+30=63^\circ$
• $\angle I = 72^\circ$

Step4: Match sides to opposite angles

Side opposite $\angle K$: $IJ$
Side opposite $\angle J$: $IK$
Side opposite $\angle I$: $JK$
(Shortest side ↔ smallest angle)

Answer:

$IJ < JK < IK$