Are the angles in a halved circle right?

Question

Imagine a circle cut in half. Now, are the two angles on it right? I can imagine when going limitedly close that they would be.

Got curious about this question after seeing this picture on the internet.

Answer 1

That depends upon how you define angle. But if you define the angle between two intersecting curves at a point $P$ at which they intersect as the angle between their tangent lines at $P$ (which is a reasonable definition) then, yes, there are two right angles in that picture.

Answer 2

If you define the angle between two vectors $\bf{x}$ and $\bf{y}$ as $$ \arccos\left(\frac{\bf{x \cdot y}}{\bf{|x||y|}}\right) \, , $$ and the angle between two intersecting curves as the angle between their tangent lines, then the angle pictured in the diagram is a right angle:

To prove this, take any two vectors that act along the tangent lines, for instance $u=-\mathbf{i}$ and $v=\mathbf{j}$. Then, $$ \mathbf{u} \cdot \mathbf{v} = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \cdot \begin{bmatrix} 0 \\ 1 \end{bmatrix} = 0 $$ and $\arccos(0)=\pi/2$. This approach to defining what an angle is requires you to define $\cos$ and $\arccos$ analytically, as otherwise the definition would be circular.