The problem gives the following information about triangle $\triangle ABC$:

• $c = 22 \, \text{cm}$ (side opposite to angle $C$),
• $m \angle B = 135^\circ$ (angle at vertex $B$),
• $a = 31 \, \text{cm}$ (side opposite to angle $A$).

This is a case where we can apply the Law of Cosines or Law of Sines to solve for missing elements (such as side $b$ or angle $A$ or $C$).

Solution Plan:

1. Find angle $C$ using the fact that the sum of angles in a triangle is $180^\circ$.
2. Use the Law of Cosines to solve for side $b$ or Law of Sines to find the other angles.

Let me calculate the missing values.

Step 1: Find angle $C$

The sum of angles in any triangle is $180^\circ$. Hence, $m \angle C = 180^\circ - 135^\circ = 45^\circ$

Step 2: Use Law of Sines to find angle $A$

We can use the Law of Sines formula: $\frac{a}{\sin A} = \frac{c}{\sin C}$ Given $a = 31 \, \text{cm}$, $c = 22 \, \text{cm}$, and $m \angle C = 45^\circ$, we solve for $\sin A$: $\frac{31}{\sin A} = \frac{22}{\sin 45^\circ}$

I'll solve this next.The measure of angle $A$ is approximately $85.12^\circ$.

Summary:

• $m \angle C = 45^\circ$,
• $m \angle A \approx 85.12^\circ$,
• $m \angle B = 135^\circ$ (given).

Now, if you'd like, we can proceed to calculate side $b$ using the Law of Cosines or finish solving the triangle using the Law of Sines.

Would you like to continue with these calculations, or do you have any other questions?

Here are some related questions:

1. How can we use the Law of Sines to find the remaining side of the triangle?
2. What happens if one of the angles is obtuse, like $m \angle B$?
3. How can we apply the Law of Cosines in this case to check our result?
4. What are other uses of the Law of Sines besides solving triangles?
5. How does this problem change if the triangle were a right triangle?

Tip: The Law of Cosines is useful when dealing with non-right triangles, especially when one of the angles is obtuse.