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. One of the triangle's angles is twice its smallest angle. What is the measure of the second largest angle of the triangle?

Introduction:
This geometry problem links the interior angles of a quadrilateral and a triangle using ratios and equality conditions. It tests your understanding of angle sums in polygons and your ability to translate ratio information into actual angle measures.

Given Data / Assumptions:

• The four angles of a quadrilateral are in the ratio 2 : 4 : 7 : 5.
• The sum of angles in a quadrilateral is 360 degrees.
• The smallest angle of the quadrilateral equals the smallest angle of a triangle.
• One angle of the triangle is twice its smallest angle.
• The sum of angles in a triangle is 180 degrees.
• We are asked to find the second largest angle of the triangle.

Concept / Approach:
First, we compute the actual angles of the quadrilateral from the ratio, then identify the smallest angle. That smallest angle becomes the smallest angle of the triangle. One triangle angle is twice this, and the third angle is found from the triangle angle-sum property. Finally, we order the triangle angles and identify the second largest.

Step-by-Step Solution:
Step 1: Find the quadrilateral angles.Let the common ratio factor be k.Angles are 2k, 4k, 7k, 5k.Sum = 2k + 4k + 7k + 5k = 18k = 360°.So k = 360° / 18 = 20°.Hence angles are 40°, 80°, 140°, 100°.Step 2: Identify the smallest angle.Smallest quadrilateral angle = 40°.This equals the smallest triangle angle.Step 3: Set up the triangle angles.Let smallest angle A = 40°.One angle B is twice A ⇒ B = 2 * 40° = 80°.Let the third angle be C.Step 4: Use triangle angle sum.A + B + C = 180° ⇒ 40° + 80° + C = 180°.C = 180° − 120° = 60°.

Verification / Alternative check:
The triangle angles are 40°, 60° and 80°. Ordered from smallest to largest: 40° (smallest), 60° (second largest), 80° (largest). All conditions are satisfied: one angle is twice the smallest, and the sum is 180°.

Why Other Options Are Wrong:
40° is the smallest, not the second largest. 80° is the largest angle, and 120° or 100° are not angles of this triangle. Only 60° correctly represents the second largest angle.

Common Pitfalls:
Errors may occur by misreading “second largest” as “second angle” or by miscomputing the quadrilateral angles. Careful stepwise calculation and then ordering of the triangle angles avoid these mistakes.

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