The angles of a triangle are in the ratio 3 : 4 : 5. A quadrilateral is formed whose three angles are equal to the three angles of this triangle. What is the sum of the largest angle and the second smallest angle of the quadrilateral?

Introduction / Context:
This question checks understanding of angle sums in triangles and quadrilaterals, along with ratios of angles. It connects two shapes by reusing the same set of three angles, and then asks for a carefully chosen sum of two angles in the quadrilateral. Handling the ratio correctly and tracking which angles are largest or smallest are crucial here.

Given Data / Assumptions:

• The three angles of the triangle are in the ratio 3 : 4 : 5.
• The sum of angles in a triangle is 180 degrees.
• A quadrilateral has a total angle sum of 360 degrees.
• Three angles of the quadrilateral are equal to the three angles of the triangle.
• We need the sum of the largest angle and the second smallest angle of the quadrilateral.

Concept / Approach:
First, find the actual angles of the triangle from the given ratio using the fact that their sum is 180 degrees. These same three angles appear in the quadrilateral. The fourth angle of the quadrilateral is obtained from the quadrilateral angle sum of 360 degrees. Once we know all four angles, we can order them from smallest to largest and add the required two angles: the largest and the second smallest.

Step-by-Step Solution:
Let the triangle angles be 3x, 4x and 5x. 3x + 4x + 5x = 12x = 180 degrees, so x = 15 degrees. Thus the angles are 3x = 45 degrees, 4x = 60 degrees and 5x = 75 degrees. The quadrilateral uses these same three angles: 45 degrees, 60 degrees and 75 degrees. Sum of these three angles is 45 + 60 + 75 = 180 degrees. Total sum of angles in a quadrilateral is 360 degrees. Therefore, the fourth angle is 360 - 180 = 180 degrees. Now the four angles are 45 degrees, 60 degrees, 75 degrees and 180 degrees. Ordered from smallest to largest: 45, 60, 75, 180. The second smallest angle is 60 degrees and the largest angle is 180 degrees. Their sum is 60 + 180 = 240 degrees.

Verification / Alternative check:
We can verify by checking the angle sums again. The triangle angles 45, 60 and 75 clearly add to 180 degrees. In the quadrilateral, including the fourth angle of 180 degrees, the total becomes 45 + 60 + 75 + 180 = 360 degrees, which matches the property of a quadrilateral. The ordering 45, 60, 75, 180 also makes sense intuitively because 180 degrees is a straight angle and clearly the largest.

Why Other Options Are Wrong:
Values like 225 degrees or 210 degrees correspond to incorrect combinations of angles or arithmetic mistakes. The options 245 degrees and 205 degrees also arise from adding the wrong pair of angles, for example the largest and the smallest or both small angles. None of those combinations match the requested “largest plus second smallest” pair.

Common Pitfalls:
Learners may mistakenly add the smallest and largest angles or the largest and second largest angles. Another error is forgetting that the quadrilateral requires a new fourth angle and simply working within the triangle. Carefully listing all four angles and ranking them prevents such confusion.

Final Answer:
The required sum of the largest angle and the second smallest angle of the quadrilateral is 240 degrees.