To solve for the measure of angle $C$ in the given triangle, we will use the Law of Cosines.

The Law of Cosines formula is:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

Where:

• $a = 50$
• $b = 90$
• $c = 55$

Solving for $\cos(C)$:

Rearrange the formula to solve for $\cos(C)$:

$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$

Let me calculate the value of $\cos(C)$, and then find angle $C$.The measure of angle $C$ is approximately $32.7^\circ$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How does the Law of Cosines differ from the Pythagorean Theorem?
2. Can the Law of Cosines be used for any type of triangle?
3. What would be the measure of the other angles, $A$ and $B$?
4. How can we verify this result using other methods?
5. How do you derive the Law of Cosines from vector mathematics?

Tip: The Law of Cosines is particularly useful for solving triangles when you know two sides and the included angle (SAS) or all three sides (SSS).