Question

determining when to use the law of cosines
which of these triangles can you use the law of cosines to solve for a missing side?

Explanation:

Step1: Recall Law of Cosines use case

The Law of Cosines ($c^2=a^2+b^2-2ab\cos(C)$) applies when we know two sides and the included angle (SAS) to find the third side, or all three sides to find an angle. We check each triangle:

Step2: Analyze first triangle

Given two angles, one side. This is AAS, use Law of Sines.

Step3: Analyze second triangle

Given two sides (12, 8) and included angle $138^\circ$ (SAS). This fits Law of Cosines for missing side $x$.

Step4: Analyze third triangle

Right triangle with one acute angle, use trig ratios or Pythagoras.

Step5: Analyze fourth triangle

Equilateral triangle, all sides equal, no need for Law of Cosines.

Answer:

The second triangle (with sides 12, 8, included angle $138^\circ$ and missing side $x$)