The triangle ABC is isosceles. What is the measure of the angle ACB? (1) angle ABC = 2 × angle ACB (2) BC < AC

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) Let’s denote each angle by its vertex, i.e. the angle BAC is the angle A, the angle ACB is the angle C and the angle ABC is the angle B.

Statement (1) is angle B = 2 × angle C . Let’s denote the measure of the angle C by x. Then angle B = 2x .

An isosceles triangle has two equal angles. From statement (1) we know for sure that the angles B and C are NOT equal. So the angle A must be equal to one of them. If angle A = angle C , then the sum of the angles in the triangle is x + x + 2x = 4x. As in any triangle the sum must be 180°. 4x = 180°. So x = 45°. In this case angle C = 45°.

However, if angle A = angle B instead, then the sum of the angles in the triangle is x + 2x + 2x = 5x. So 5x = 180° and x = 36°. In this case angle C = 36°. We have two different possible values of the angle C, so statement (1) by itself is NOT sufficient.

Statement (2) defines BC < AC. In any triangle, the greater angle is subtended by the greater side. So we can say that angle B > angle A. But we know no measures of the angles and we know nothing about the angle C. Therefore statement (2) by itself is NOT sufficient.

If we use the two statements together, statement (2) will imply that angle B > angle A. So we will have only one option left in statement (1): angle C = angle A . As we know, this yields angle C = 45° . Therefore the both statements taken together are sufficient.

Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C. ---------- May I suggest that you insert diagrams for triangle questions because it would greatly assist in the comprehension of (i) the information given and (ii) the question asked. Alternatively state which letter corresponds to the triangle's top vertex.

Alternatively state which letter corresponds to the triangle's top vertex.

That is what this question is about. You are given some arbitrary isosceles triangle and some additional information about it. You should figure out on your own which sides/angles are equal (or what are the possible options).

As you can see we easily named the angles with the respective vertices in the explanation. It helped us to comprehend and deal with the information along the way.

Here are the possible options:

Statement (1):

The triangle can be a 45°-90°-45° triangle.

The triangle can be a 72°-36°-72° triangle.

Statement (2) gives way too many options.

When the statements are combined, we are left with a 45°-90°-45° triangle only.

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