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16 Sep 2014, 01:19
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Are all the angles in triangle ABC less than 90 degrees?
(1) \(AB < BC < AC\).
(2) \(\angle ABC = 80\) degrees.
(1) \(AB < BC < AC\).
(2) \(\angle ABC = 80\) degrees.
16 Sep 2014, 01:19
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Official Solution:
Are all the angles in triangle ABC less than 90 degrees?
(1) \(AB < BC < AC\)/
Knowing that the lengths of all three sides are different is not sufficient to answer the question.
(2) \(\angle ABC = 80\) degrees.
This information alone is also insufficient. For example, consider the angles to be 80°, 80°, and 20°, in which case all the angles are less than 90 degrees. Alternatively, consider the angles to be 80°, 95°, and 5°, in which case not all the angles are less than 90 degrees.
(1)+(2) The first statement indicates that AC is the longest side of the triangle. Since the largest angle in a triangle is opposite the longest side, \(\angle ABC\) is the largest angle. From statement (2), we know that the largest angle is 80°, which means that all angles in the triangle are less than 90°. Sufficient.
Answer: C
Are all the angles in triangle ABC less than 90 degrees?
(1) \(AB < BC < AC\)/
Knowing that the lengths of all three sides are different is not sufficient to answer the question.
(2) \(\angle ABC = 80\) degrees.
This information alone is also insufficient. For example, consider the angles to be 80°, 80°, and 20°, in which case all the angles are less than 90 degrees. Alternatively, consider the angles to be 80°, 95°, and 5°, in which case not all the angles are less than 90 degrees.
(1)+(2) The first statement indicates that AC is the longest side of the triangle. Since the largest angle in a triangle is opposite the longest side, \(\angle ABC\) is the largest angle. From statement (2), we know that the largest angle is 80°, which means that all angles in the triangle are less than 90°. Sufficient.
Answer: C
02 May 2023, 10:08
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
