If angles of a quadrilateral are in the ratio 3:5:9:13, find the largest angle ?

Concept: The sum of the interior angles of any quadrilateral is always 360°. When the angles are given in a ratio, assign each ratio term a common multiplier x, equate their sum to 360°, and solve for x to get each individual angle.

Application:

1. Let the four angles be 3x, 5x, 9x and 13x, matching the given ratio 3 : 5 : 9 : 13.

2. Since the angles of a quadrilateral sum to 360°: 3x + 5x + 9x + 13x = 360°.

3. Adding the coefficients: 30x = 360°.

4. Solve for x: x = 360° ÷ 30 = 12°.

5. The largest angle corresponds to the highest ratio term, 13x: largest angle = 13 × 12° = 156°.

Cross-check:

Because the ratio terms 3, 5, 9, 13 must be multiplied by the same x, the largest angle has to be an exact multiple of 13. Dividing 156 by 13 gives exactly 12, confirming the computed value is consistent.