Question

How many obtuse angles can an equilateral triangle have?

Answer 1

None. An equilateral triangle has three equal sides, and thus three equal angles.

We can prove this using the law of cosines with the SSS case.

`a = b = c`

`c^2 = a^2 + b^2 - 2abcos/_C`

becomes

`a^2 = a^2 + a^2 - 2a*a*cos/_A`

`-a^2 = -2a^2cos/_A`

`1 = 2cos/_A`

`1/2 = cos/_A`

`color(blue)(/_A = 60^@)`

Since only one side `a` corresponds to only one `/_A`, and since sides `a = b = c`, we have `/_A = /_B = /_C`.

Acute angles cannot be greater than `90^@`, and obtuse angles cannot be less than `90^@`. All triangles can only have `theta_"tot" = 180^@`.

Therefore, with three acute angles, there are no obtuse angles.