A triangle has angles in the ratio 2:3:4. What are the measures of the three angles?

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We have a triangle with angles in the ratio 2 to 3 to 4. Let's call these angles 2x, 3x, and 4x. We need to find the actual measures of these three angles. Since the sum of angles in any triangle equals 180 degrees, we can set up the equation: 2x plus 3x plus 4x equals 180 degrees. Combining like terms, we get 9x equals 180 degrees. Now we solve for x by dividing both sides by 9. Nine x equals 180 degrees, so x equals 180 divided by 9, which gives us x equals 20 degrees. Now we can find each angle. The first angle is 2 times 20 degrees, which equals 40 degrees. The second angle is 3 times 20 degrees, which equals 60 degrees. The third angle is 4 times 20 degrees, which equals 80 degrees. Let's verify our answer. First, we check that the angles sum to 180 degrees: 40 plus 60 plus 80 equals 180 degrees. Second, we verify the ratio: 40 to 60 to 80 simplifies to 2 to 3 to 4. Both conditions are satisfied, so our answer is correct. The three angles are 40 degrees, 60 degrees, and 80 degrees.