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.

The question asks if AD = BD. This is really important here.

We do not assume it. We've proven it, based on the fact from statement (1).

An isosceles triangle is a triangle, two of which sides are equal. The angles opposing to these sides are equal as well.

But what if we don't know if some particular triangle is isosceles or not? In this case we need to know just one of the two properties, i.e.:
two sides are equal -> the triangle is isosceles -> the two angles are equal
or
two angles are equal -> the triangle is isosceles -> the two sides are equal