Which of the following sets could be the measures of the three interior angles of a triangle?

Introduction / Context:
This question checks whether you know the basic angle sum property of a triangle and can quickly verify if a set of three angles can be internal angles of a triangle. It is a simple but very common test of fundamental geometry knowledge.

Given Data / Assumptions:

• Each option lists three angles in degrees.
• We assume these are interior angles of a triangle, if possible.
• A valid triangle must have all interior angles positive and their sum equal to 180°.

Concept / Approach:
The interior angles of any triangle in Euclidean geometry always add up to 180°. Therefore, to check if a set of three angles can represent the angles of a triangle, we simply add them and verify that the total is exactly 180° and that none of the angles is zero or negative. If the sum is greater than or less than 180°, or one angle is zero, the set cannot represent a triangle.

Step-by-Step Solution:
Step 1: Option a: 33°, 42°, 115°. Sum = 33° + 42° + 115° = 190°, which is greater than 180°, so this is not possible. Step 2: Option b: 40°, 70°, 80°. Sum = 40° + 70° + 80° = 190°, again greater than 180°, so not possible. Step 3: Option c: 30°, 60°, 100°. Sum = 30° + 60° + 100° = 190°, also greater than 180°, so this set is impossible as well. Step 4: Option d: 50°, 60°, 70°. Sum = 50° + 60° + 70° = 180°, which exactly matches the triangle angle sum. Step 5: All angles in option d are positive and less than 180°, so this set is valid for a triangle.

Verification / Alternative Check:
Because only one option yields a sum of exactly 180°, there is no ambiguity. Any small change, such as increasing or decreasing one angle while keeping the others fixed, would break the 180° total, so only 50°, 60°, and 70° can work together as the internal angles of a triangle.

Why Other Options Are Wrong:
Options a, b, and c all sum to 190°, which is too large. A triangle with such angles would have to "overlap" itself, which is impossible in flat Euclidean geometry. The angle sum property is strict: the sum must be exactly 180° for a standard triangle in a plane.

Common Pitfalls:
Learners sometimes assume that any three positive angles less than 180° must form a triangle without checking the sum. Another mistake is misadding the angles in haste, especially under exam pressure. Taking a few seconds to carefully sum the values and compare with 180° is enough to avoid such errors.

Final Answer:
The only possible set of angles of a triangle is 50°, 60°, 70°.