1. ## Circle geometric reasoning.

ABCO is a quadrilateral. ABC lies on the circumference of a circle and O is on the centre of the circle. Angle BAC is 39 degrees. Angle BCA is labeled x. Angle ACO is labeled y.

Use geometric reasoning to find the unknown angle y, in terms of x.

Image:

Here's what I got:

Angle ABC = 141 - x (Angles in a triangle must add to 180)
Reflex angle COA = 282 -2x (Angle at the centre is double the angle at the circumference)
Angle COA = 78 + 2x (Angles on a point must add to 360)

This is where I get stuck. I cannot get any other angles. Is there a really simple geometric proof that I'm missing?

Point D was drawn there on the original question as well. Maybe it is used, or maybe it is just there to distract you.

2. Hm... angle AOB is equal to 2x, since angle at centre is double angle at circumference.

This gives angle COB = 78

Now I'm not sure where to go... give me some time to think about it.

3. Ah, I missed the obvious. Angle CAO is equal to angle ACO since the triangle OAC is an isoscelese traiangle with sides OC and OA equal since they are radii of the circle.

From there, we get:

$y + y + 78 +2x = 180$

It should be okay now.

4. Ah, how did I not see that! That's so obvious.

Thanks for helping.