In triangle ABC, if 3 times angle A is equal to 4 times angle B, which is equal to 6 times angle C, how can we calculate the angles of the triangle?

To find the angles of triangle ABC given the relationships between the angles, let’s denote the measures of the angles as follows:

According to the problem, we have the following relationships:

We can express angles B and C in terms of angle A:

Now, the sum of the angles in any triangle is always 180 degrees:

Substituting the expressions for B and C in terms of A into the equation:

To combine these, we first need a common denominator. The common denominator for 4 and 2 is 4, so we can rewrite (1/2)A as (2/4)A:

Now combining the terms gives us:

Now, we can solve for angle A:

Now that we know angle A, we can find angles B and C:

To summarize, the angles of triangle ABC are:

These angles satisfy the given relationships and add up to 180 degrees, confirming our calculations are correct.