The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. What is the…

Concept

The sum of the interior angles of any quadrilateral is always 360°. This follows from the general polygon angle-sum rule: for a polygon with n sides, the sum of interior angles = (n − 2) × 180°; for n = 4 this gives (4 − 2) × 180° = 360°. When a quadrilateral's angles are given in a ratio, each ratio unit stands for a fixed number of degrees — found by dividing the total angle sum by the sum of the ratio parts.

Step-by-Step Solution

1. Add the ratio parts: 3 + 5 + 7 + 9 = 24.

2. Divide the total angle sum by the number of parts to find the value of one part: 360° ÷ 24 = 15°.

3. Multiply each ratio term by 15° to get the four angles: 3 × 15° = 45°, 5 × 15° = 75°, 7 × 15° = 105°, 9 × 15° = 135°.

4. Identify the least angle (45°, from the smallest ratio term 3) and the greatest angle (135°, from the largest ratio term 9).

5. Subtract to get the required difference: 135° − 45° = 90°.

Cross-Check

The four angles must add back to 360°: 45° + 75° + 105° + 135° = 360° ✓. Also, since every ratio unit is worth the same 15°, the gap between the least and greatest angles can be found directly from the gap between their ratio terms: (9 − 3) × 15° = 6 × 15° = 90°, matching the step-by-step result.

So the difference between the least and the greatest angles of the quadrilateral is 90°.