The measures of the four interior angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. What is the measure, in degrees, of the largest angle of the quadrilateral?

Introduction / Context:
This question tests your understanding of angle sums in polygons and how to use given ratios to find individual angle measures. A quadrilateral has four interior angles whose sum is fixed. When the angles are in a given ratio, you can treat them as multiples of a common factor, solve for that factor, and then determine each angle, including the largest one.

Given Data / Assumptions:

• The quadrilateral has four interior angles.
• The ratio of the angles is 3 : 4 : 5 : 6.
• Sum of interior angles of a quadrilateral is 360 degrees.
• Let the common factor be k, so angles are 3k, 4k, 5k and 6k.

Concept / Approach:
For any quadrilateral in Euclidean geometry, the sum of the four interior angles equals 360 degrees. When the angles are given in a ratio, you express each angle as a multiple of a common unknown k. The sum of these expressions must equal 360 degrees, giving a simple linear equation in k. Once k is known, each angle can be computed. The largest angle corresponds to the largest ratio term, which in this case is 6k.

Step-by-Step Solution:
Let the angles be 3k, 4k, 5k and 6k degrees. Sum of interior angles of a quadrilateral is 360 degrees. So 3k + 4k + 5k + 6k = 360. Add the coefficients: (3 + 4 + 5 + 6)k = 18k. Thus 18k = 360. k = 360 / 18 = 20 degrees. Angles are therefore 3k = 60°, 4k = 80°, 5k = 100° and 6k = 120°. The largest angle is 6k = 120°.

Verification / Alternative check:
Check that the calculated angles sum correctly: 60° + 80° + 100° + 120° = 360°, which matches the known total for a quadrilateral. The ratio 3 : 4 : 5 : 6 is preserved since 60 : 80 : 100 : 120 simplifies to 3 : 4 : 5 : 6 when divided by the common factor 20. This confirms both the ratio and the total angle sum, and thus the largest angle of 120 degrees is consistent.

Why Other Options Are Wrong:
100°: This corresponds to the third largest angle (5k), not the largest one. 80° and 60°: These are the smaller angles 4k and 3k respectively. 140°: This angle does not appear in the set and would break the given ratio and total sum of 360 degrees.

Common Pitfalls:
Forgetting that the angle sum of a quadrilateral is 360 degrees, not 180 degrees as in a triangle. Adding the ratio numbers incorrectly or miscomputing 360 / 18. Choosing 100 degrees because it looks large without checking that 120 degrees is actually larger and valid.

Final Answer:
The measure of the largest angle is 120°