Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it

Prove by contradiction that in a triangle, at least one of the angles is not less than 60∘60^{\circ}. The correct assumption to make is (　　)
A: Assume all three angles are not greater than 60∘60^{\circ}
B: Assume all three angles are less than 60∘60^{\circ}
C: Assume at most one angle is greater than 60∘60^{\circ}
D: Assume at most two angles are less than 60∘60^{\circ}

Solution

To prove the proposition: "In a triangle, at least one angle is not less than 60∘60^{\circ}" using the method of contradiction, since this proposition is a particular proposition, we should assume: "All three angles in the triangle are less than 60∘60^{\circ}".

Therefore, the correct choice is: B\boxed{B}

Since the proposition given in this problem is a particular proposition, its negation is the condition for the assumption. By writing out its negation and comparing it with the four options, we can find the answer.

This problem examines the basic concept of the method of contradiction. The key to solving it is to understand the rules of the method of contradiction and that the negation of a particular proposition is a universal proposition. This problem is a basic concept examination question, so it is important to remember and understand it.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.