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Question

Two of a triangle's interior angles measure 4545^{\circ} and 5555^{\circ}. If the triangle's side lengths are represented by a, b, and c and a<b<c, which of the following statements is true for this triangle? A. a2+b2>c2a^{2}+b^{2}>c^{2}, B. a2+b2<c2a^{2}+b^{2}<c^{2}, C. a2+b2=c2a^{2}+b^{2}=c^{2}, D. Not enough information to determine.

Solution

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By Triangle Sum Theorem, the missing angle measures 180°45°55°=80°180\text{\textdegree}-45\text{\textdegree}-55\text{\textdegree} =80\text{\textdegree}. Since all angles are acute, then the triangle is an acute triangle.

By the Pythagorean acute inequality theorem, a triangle with side lengths aa, bb, and cc where cc is the longest side form an acute triangle if:

a2+b2>c2a^2+b^2>c^2

So, the correct answer is choice A.\textbf{\color{#c34632}A.}

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