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  3. In quadrilateral ABCD, is angle BCD a right angle?(1) Angle ABC is a right angl

In quadrilateral ABCD, is angle BCD a right angle?(1) Angle ABC is a right angl

游客2025-02-27  2
问题 In quadrilateral ABCD, is angle BCD a right angle?
(1) Angle ABC is a right angle.
(2) Angle ADC is a right angle.
选项 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.
答案 E
解析 (1)    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.

(2)    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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