To solve the triangle with sides \( a = 14 \), \( b = 21 \), and \( c = 19 \), we need to find its angles and other missing parameters.

To solve the given triangle with sides a=14, b=21, and c=19, we'll use the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. Before we start, we should check if the triangle is valid.

A triangle is valid if the sum of any two sides is greater than the third side. In this case: 14 + 21 > 19 (which is 35 > 19), 14 + 19 > 21 (which is 33 > 21), and 21 + 19 > 14 (which is 40 > 14). Since all these conditions are true, we have a valid triangle.

Now, let's use the Law of Cosines to find the angles of the triangle. The Law of Cosines states: c^2 = a^2 + b^2 - 2ab*cos(C) b^2 = a^2 + c^2 - 2ac*cos(B) a^2 = b^2 + c^2 - 2bc*cos(A)

We can use any of these formulas to find one angle, and then proceed to find the others.

Let's find angle C first: 19^2 = 14^2 + 21^2 - 2 * 14 * 21 * cos(C) 361 = 196 + 441 - 588 * cos(C) 361 = 637 - 588 * cos(C)

Now solve for cos(C): 588 * cos(C) = 637 - 361 588 * cos(C) = 276 cos(C) = 276 / 588 cos(C) = 0.4694 (approximately)

Using a calculator to find the angle from the cosine value: C = cos^(-1)(0.4694) (cosine inverse or arccosine) C ≈ 62.05° (approximately)

Next, let's find angle A using the Law of Cosines: 14^2 = 21^2 + 19^2 - 2 * 21 * 19 * cos(A) 196 = 441 + 361 - 798 * cos(A)

Now solve for cos(A): 798 * cos(A) = 441 + 361 - 196 798 * cos(A) = 802 - 196 798 * cos(A) = 606 cos(A) = 606 / 798 cos(A) = 0.7594 (approximately)

Using a calculator to find the angle from the cosine value: A = cos^(-1)(0.7594) A ≈ 40.54° (approximately)

Now that we have two angles, we can find the third angle using the fact that the sum of angles in a triangle is 180°. Let's find angle B: B = 180° - A - C B = 180° - 40.54° - 62.05° B ≈ 77.41° (approximately)

So, the measures of the angles of the triangle are approximately: A ≈ 40.54°, B ≈ 77.41°, and C ≈ 62.05°.