Final answer:
To find the interior angles of the triangle with the given vertices, we can use vectors. The angle between vectors AB and BC is 90 degrees. The remaining two angles are approximately 45.64 degrees each.
Step-by-step explanation:
To find the interior angles of the triangle with the given vertices, we can use vectors. Let's label the vertices as A(4, 5), B(13, 14), and C(12, 15). We first need to find the vectors AB and BC. Vector AB is obtained by subtracting the coordinates of point A from point B: AB = (13-4, 14-5) = (9, 9). Vector BC is obtained by subtracting the coordinates of point B from point C: BC = (12-13, 15-14) = (-1, 1).
Next, we can find the angle between vectors AB and BC using the dot product formula: angle = arccos((AB·BC) / (|AB| * |BC|)). The dot product of AB and BC is AB·BC = (9 * -1) + (9 * 1) = 0. The magnitude of vector AB is |AB| = sqrt((9^2) + (9^2)) = sqrt(162) ≈ 12.73, and the magnitude of vector BC is |BC| = sqrt((-1^2) + (1^2)) = sqrt(2).
Substituting the values into the formula, angle = arccos(0 / (12.73 * sqrt(2))). Using a calculator, the angle is approximately 90 degrees.
Since the sum of the interior angles of a triangle is always 180 degrees, we can find the other two angles by subtracting the known angle from 180. Therefore, the remaining two angles are approximately 45.64 degrees each.