Triangle inscribed in circle, vertex at circle's center, solve for unknown angles.

Question

$O$ is the center of the circle , $A$ and $B$ lie on the circle

what are the possible values of $x$ and $y$

I found answers options , asked to mark one or more

A) X = 80 Degree AND Y = 50 Degree
B) X = 50 Degree AND Y = 80 Degree
C) X = 450 Degree AND Y = 90 Degree
D) X = 700 Degree AND Y = 30 Degree
E) X = 60  Degree AND Y = 60 Degree

Answer 1

We know

• $(1)$: the sum of the angles equals $180^\circ$.

• $(2)$: Two sides of $\triangle AOB$, namely $\overline{OA}, \overline{OB}$ have equal length. Why? Because $\;|\overline{OA}| = |\overline{OB}| = r$, the length of the radius of the circle.

So from $(2)$ we know that $\triangle AOB$ is an isosceles triangle, and that the angles opposite sides $\overline{OA}, \overline{OB}$ are necessarily equal in measure. This means that the unmarked angle has measure equal to $x$.

From $(1):$ Summing the angles gives us $\;x + x + y = 2x + y = 180\tag{3}$

Substituting each pair of values $(x, y)$ into equation $(3)$ rules out all choices except for $(B)$ and $(E)$: Both $(B)$ and $(E)$ are valid choices.

(Note, for the very large x values which are greater than $180$, we can compute their equivalent angles by subtracting $180$ from $x$ until $x' \lt 180$. Then evaluate $(3)$ using $x', y$. It turns out that in both cases where the given $x > 180$, the resulting $x'$s with corresponding y values fails to satisfy $(3)$

Answer 2

Only valid answers are a) b) and e) but from figure you can see that not all angles are the same, so corect are a) and b). It's because its circle, so triangle have 2 equal sides(r) so 2 angles have to be the same(x and (180-y-x)). So your answer must satisty equation $2*x + y = 180$.

Answer 3

We know that our given triangle is (at least) isosceles by the fact that two of the legs are radii of the circle and therefore the same length. Thus, if we label the last angle $z$, then $x=z$.

From here, as Naimads said, you're trying to satisfy the equation $2x+y=180$.

At this point, it's easy to check (and eliminate several) answers and find out which solutions are correct.