Question

Suppose a triangle has sides a, b, and c, and that a^2 + b^2 > c^2. Let θ be the measure of the angle opposite the side of length c. Which of the following must be true? Check all that apply.
A. The triangle in question is a right triangle.
B. cosθ > 0
C. cos θ< 0
D. θ is an acute angle.

Answer 1

By using al-Kashi's theorem
\[c^2=a^2+b^2-2ab \cos(\theta)\]
\[a^2+b^2>c^2 \iff -2ab \cos(\theta) >0 \iff 2ab \cos(\theta) < 0 \iff \cos(\theta)<0\]

This means the angle must be greater than a right angle... so an obtuse angle.