题目信息
In quadrilateral ABCD, is angle BCD a right angle?
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B:Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D:EACH statement ALONE is sufficient.
E:Statements (1) and (2) TOGETHER are NOT sufficient.
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已选答案：
正确答案：
E:Statements (1) and (2) TOGETHER are NOT sufficient.
Geometry Quadrilaterals
The figure below shows two possibilities for a quadrilateral ABCD such that angle ABC is a right angle. One quadrilateral is such that angle BCD is not a right angle (i.e., the answer to the question can be NO), and the other quadrilateral is a square (i.e., the answer to the question can be YES); NOT sufficient.
The figure below shows that relabeling the vertices of the examples used in (1) above will give an example such that angle BCD is not a right angle and an example such that angle BCD is a right angle; NOT sufficient.
Given (1) and (2), the figure below indicates how an example satisfying both (1) and (2) can be constructed such that angle BCD is not a right angle. A right angle is constructed with vertex B and another right angle, appropriately rotated with respect to the first right angle, is constructed with vertex D. The rays of these two angles intersect at points A and C to form a quadrilateral ABCD that satisfies both (1) and (2) and is such that angle BCD is not a right angle (i.e., the answer to the question can be NO). For completeness, a square is also shown, which satisfies both (1) and (2) and is such that angle BCD is a right angle (i.e., the answer to the question can be YES).
The correct answer is E;both statements together are still not sufficient.
The figure below shows two possibilities for a quadrilateral ABCD such that angle ABC is a right angle. One quadrilateral is such that angle BCD is not a right angle (i.e., the answer to the question can be NO), and the other quadrilateral is a square (i.e., the answer to the question can be YES); NOT sufficient.
The figure below shows that relabeling the vertices of the examples used in (1) above will give an example such that angle BCD is not a right angle and an example such that angle BCD is a right angle; NOT sufficient.
Given (1) and (2), the figure below indicates how an example satisfying both (1) and (2) can be constructed such that angle BCD is not a right angle. A right angle is constructed with vertex B and another right angle, appropriately rotated with respect to the first right angle, is constructed with vertex D. The rays of these two angles intersect at points A and C to form a quadrilateral ABCD that satisfies both (1) and (2) and is such that angle BCD is not a right angle (i.e., the answer to the question can be NO). For completeness, a square is also shown, which satisfies both (1) and (2) and is such that angle BCD is a right angle (i.e., the answer to the question can be YES).
The correct answer is E;both statements together are still not sufficient.
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