The measure of angle E, calculated using the Law of Cosines, is approximately 61.26 degrees.
To find the measure of angle E, we can use the Law of Cosines, which states that for any triangle with sides a, b, and c and opposite angles A, B, and C, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
We can apply this formula to the given triangle, where side c is 13, side a is 10, and side b is 12. We want to find angle C, which is angle E in the triangle. Plugging in the values, we get:
12^2 = 10^2 + 13^2 - 2(10)(13) * cos(E)
144 = 100 + 169 - 260 * cos(E)
Now, combine like terms:
144 = 269 - 260 * cos(E)
Subtract 269 from both sides:
-125 = -260 * cos(E)
Divide by -260:
cos(E) = 125/260
cos^-1(125/260) = 61.264