O is the centre of a circle
A, B and C are points on this circle
angle COA is 130 degrees
Find the size of CBA
I have no idea how to approach this question, answer is not necessary, merely how to approach it.
it depends on whether or not B is on the short arc between A and C
or on the longer arc.
In both cases, you can use.....
the angle at the centre of the circle is twice the angle at the circumference, for angles standing on the same arc.
Hence angle CBA is found from angle COA.
the arcs are the curved line segments of the circumference of the circle.
The longer arc in the attachment goes from C around the circle to the left
until reaching A.
If B is on that side, then angle ABC is half of 130 degrees.
However if B is on the shorter arc, then angle ABC is half of 230 degrees.
The blue angles are the angles at the circumference,
the black ones are the angles at the centre.
then that angle is one-half the measure
of its intercepted arc.
What is the measre of its intercept arc, you ask?
Well, it's the same as the measure of the central angle that also intercepted it.
That central angle is angle COA which you originally stated as being 130 degrees.
Therefore, angle CBA = 1/2(130) = 65 degrees.
Remember this theorem: The measure of an inscribed angle in a circle is equal to one-half the measure of its intercepted arc.