The angles of a quadrilateral are in the ratio 2 : 4 : 7 : 5. The smallest angle of this quadrilateral is equal to the smallest angle of a triangle. In that triangle, one angle is twice the smallest angle. What is the second largest angle of the triangle (in degrees)?

Introduction / Context:
This question links the internal angles of a quadrilateral with those of a triangle. It first gives the ratio of angles in a quadrilateral and then uses the smallest of these angles as the smallest angle of a triangle. In that triangle, another angle is twice the smallest angle. The task is to identify the second largest angle of the triangle. This combines knowledge about angle sums in polygons and basic algebraic manipulation of ratios.

Given Data / Assumptions:

• The four angles of a quadrilateral are in the ratio 2 : 4 : 7 : 5.
• The sum of the interior angles of a quadrilateral is 360 degrees.
• The smallest angle of the quadrilateral equals the smallest angle of a triangle.
• One of the angles of that triangle is twice its smallest angle.
• The sum of interior angles of a triangle is 180 degrees.

Concept / Approach:
First, convert the ratio of the quadrilateral angles into actual angle measures. Let the common factor be k. Then the four angles are 2k, 4k, 7k and 5k. Their sum must equal 360 degrees, which allows us to find k and thus each angle. The smallest angle of the quadrilateral will be the smallest angle of the triangle. In the triangle, if the smallest angle is x degrees, then another angle is 2x degrees. The third angle can be found because the sum of angles in a triangle is 180 degrees. Finally, we order the three triangle angles and identify the second largest one.

Step-by-Step Solution:
Let the quadrilateral angles be 2k, 4k, 7k and 5k. Sum of angles of a quadrilateral = 360 degrees. So 2k + 4k + 7k + 5k = 18k = 360 ⇒ k = 20. Thus the four angles are 40°, 80°, 140° and 100°. The smallest angle is 40°, so the smallest angle of the triangle is also 40°. Let the triangle angles be 40°, 2 × 40° = 80°, and the third angle C. Sum of angles of triangle = 180°, so C = 180 − 40 − 80 = 60°. The three triangle angles are 40°, 60° and 80°; ordered from smallest to largest they are 40°, 60°, 80°. Therefore, the second largest angle is 60°.

Verification / Alternative check:
We can double check that 40° truly is the smallest quadrilateral angle, which it is compared to 80°, 100° and 140°. In the triangle, 40° and 80° satisfy the condition that one angle is twice the smallest. The remaining angle 60° completes the total of 180°. The largest triangle angle is 80°, the second largest is 60°, and the smallest is 40°, so the answer is consistent with all conditions.

Why Other Options Are Wrong:

• 80 degrees: This is the largest triangle angle, not the second largest.
• 120 degrees: No triangle angle here is 120°, and such a large angle would violate the given conditions.
• 40 degrees: This is the smallest angle, not the second largest.
• 100 degrees: This comes from the quadrilateral but does not appear as a triangle angle.

Common Pitfalls:
A common mistake is to misidentify the smallest quadrilateral angle or to forget that triangle angles must sum to 180°. Another error is to assume that the second largest triangle angle must come directly from the quadrilateral without recalculating. Carefully computing all angles step by step avoids these errors and clarifies which angle is truly the second largest.

Final Answer:
The second largest angle of the triangle is 60 degrees.