According to the figure above, does

*AD* =

*BD*?

(1) The degree measure of angle

*ABD* is half that of angle

*BDC*.

(2) The degree measure of angle

*DBC* is 20°.

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.

(A) Using Statement (1), let's call angle

*BDC* =

*x*. Now, angle

*BDA* = 180° –

*x* and angle

*ABD* =

*x*/2.

Furthermore, all three angles in triangle

*ABD* must sum to 180°, so we can write the following equation:

180° =

*x*/2 + (180° –

*x*) + angle

*BAD*.

Solving this equation, we find that angle

*BAD* =

*x*/2, which is the same degree measure as that of angle

*ABD*. Since these two angles are the same, the triangle is isosceles, and

*AD* =

*BD*. Statement (1) is sufficient.

Using Statement (2) alone, we cannot determine any relationships between the other angles, so Statement (2) is insufficient.

Since Statement (1) is sufficient and Statement (2) is not, the correct answer is choice (A).

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I don't get how Statement 1 is correct. The question asks if AB = BD. But in the answer it assumes that angle ABD = angle BAD. I thought you can assume unless it is given as a fact/statement.