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In geometry, a triangle is a closed figure with three straight sides and three angles. The sum of the interior angles of a triangle is always 180 degrees. Therefore, not all sets of angles can form a triangle. For a set of angles to form a triangle, they must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We will discuss which set of angles can form a triangle.

Let's assume that we have three angles, A, B, and C. The sum of these angles is 180 degrees. Therefore, we can write the following equation:

A + B + C = 180 degrees

We know that for a set of angles to form a triangle, the sum of any two angles must be greater than the third angle. Therefore, we can write the following three inequalities:

A + B > C

B + C > A

A + C > B

B + C > A

A + C > B

Let's consider the three angles A, B, and C separately and see what ranges of values they can take to satisfy the triangle inequality theorem.

Angle A:

Let's assume that angle A is the largest angle among the three. In this case, the other two angles must be less than A. Therefore, we can write the following inequality:

A > B, A > C

Now, let's consider the two smaller angles, B and C. For them to form a triangle with A, their sum must be greater than A. Therefore, we can write the following inequality:

B + C > A

By combining these two inequalities, we get:

A > B, A > C, B + C > A

Therefore, any value of angle A that satisfies these three inequalities can form a triangle.

Angle B:

Now let's assume that angle B is the largest angle among the three. In this case, the other two angles must be less than B. Therefore, we can write the following inequality:

B > A, B > C

Now, let's consider the two smaller angles, A and C. For them to form a triangle with B, their sum must be greater than B. Therefore, we can write the following inequality:

A + C > B

By combining these two inequalities, we get:

B > A, B > C, A + C > B

Therefore, any value of angle B that satisfies these three inequalities can form a triangle.

Angle C:

Finally, let's assume that angle C is the largest angle among the three. In this case, the other two angles must be less than C. Therefore, we can write the following inequality:

C > A, C > B

Now, let's consider the two smaller angles, A and B. For them to form a triangle with C, their sum must be greater than C. Therefore, we can write the following inequality:

A + B > C

By combining these two inequalities, we get:

C > A, C > B, A + B > C

Therefore, any value of angle C that satisfies these three inequalities can form a triangle.

For a set of angles to form a triangle, the sum of any two angles must be greater than the third angle. Therefore, we can use the triangle inequality theorem to determine which set of angles can form a triangle. Specifically, for any given set of three angles A, B, and C, we can determine which angle is the largest and then use the corresponding inequality to determine the range of values for that angle that can form a triangle. If all three inequalities are satisfied, then the set of angles can form a triangle.

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