The angles of a triangle are in the ratio 3:7:8. Which is the greatest angle?

The interior angles of any triangle always add up to 180°. When the angles are given as a ratio, each angle can be written as its ratio part multiplied by a common value x, and x is found by dividing the total angle sum (180°) by the sum of the ratio parts.

1. Let the three angles be 3x, 7x, and 8x, based on the given ratio 3:7:8.

2. Since the angles of a triangle sum to 180°, form the equation 3x + 7x + 8x = 180°.

3. Combine the like terms on the left side: 18x = 180°.

4. Solve for x by dividing both sides by 18: x = 180° ÷ 18 = 10°.

5. Compute each angle using this value of x: 3x = 30°, 7x = 70°, and 8x = 80°.

6. The greatest angle corresponds to the largest ratio part (8), so the greatest angle is 80°.

Cross-check: 30° + 70° + 80° = 180°, which confirms the angle sum property, and 80° is indeed the largest of the three computed angles.

Therefore, the greatest angle of the triangle is 80°.