
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.