In the figure O is the centre of the circle and angle BOC=130. Find the measure of angle ADC​

To solve this problem, we need to understand a few properties of circles:

1. The angle at the center of a circle is twice the angle at the circumference when the angles subtend the same arc. This is also known as the Angle at the Center Theorem.

Given that angle BOC = 130 degrees, and angles BOC and BAC subtend the same arc BC, we can find angle BAC by dividing the central angle by 2.

So, angle BAC = BOC/2 = 130/2 = 65 degrees.

2. Angles on a straight line add up to 180 degrees.

Given that angle BAC = 65 degrees, and angles BAC and CAD are on the same straight line, we can find angle CAD by subtracting angle BAC from 180.

So, angle CAD = 180 - BAC = 180 - 65 = 115 degrees.

3. Angles in a triangle add up to 180 degrees.

Given that angle CAD = 115 degrees, and angles CAD, ADC, and CDA form a triangle, we can find angle ADC by subtracting the sum of angles CAD and CDA (which is equal to angle BAC) from 180.

So, angle ADC = 180 - (CAD + CDA) = 180 - (115 + 65) = 0 degrees.

Therefore, the measure of angle ADC is 0 degrees.