We have seen that the opposite angles of a cyclic quadrilateral add up to 180 degrees. A quadrilateral is cyclic when a circle passes through all its four vertices. Of course it’s an important property but we need to prove this result. We are given that quadrilateral ABCD is cyclic. And we have to prove that it’s opposite angles add up to 180 degrees. This would mean angle A plus angle C equals 180 degrees. And angle B plus angle D equals 180 degrees. The figure we have is absolutely blank. Nothing is given to us about the sides or the angles. The only thing we do know is that all the four vertices ABC and D lie on the circle. Since the figure tells us nothing, let’s try and analyze what we have to prove first. We have to prove that the sum of angles is 180 degrees. Looking at this 180 degrees, we would probably have to form a triangle somewhere in the figure and use the sum of Interior angles concept. If we join any one diagonal, we get two triangles. But these two triangles also don’t tell us anything. So in our construction let us join both the diagonals AC and BD. AC and BD. That gives us four triangles, but we won’t use all of them. More than triangles, what this gives us is angles in the same segment. We will understand it in the proof. There are four chords AB, BC, CD and DA in this circle and each one divides the circle into two segments. Now understand this really well, the colored region is a major segment formed by chord AB. And angle ACB and angle ADB lie in that segment. Since angles formed in the same segment are equal, we can say that angle ACB is equal to angle ADB. Now look at this blue region. It is a major segment formed by chord BC and it contains two angles, angle BAC and angle BDC. So, we can say that angle BAC is equal to angle BDC. Both these pairs are angles formed in the same segment. Look at the figure now. If we add angle ADB and angle BDC, we will get angle ADC which is actually angle D. So if we add the right hand sides of these two equations, we get angle D But we can’t just add the right hand sides, we will have to add the left hand sides too. Adding these two equations, We get angle ACB plus angle BAC equals angle D. It gets even more interesting. Since we are looking for 180 degrees, we have to look for sum of angles of a triangle. Look at the left-hand side. It’s angle ACB plus angle BAC. This is angle ACB and this is angle BAC. If we look at triangle ACB the only angle left out is angle ABC. If we add angle ABC to the left hand side, we will get the sum of all the angles of a triangle ACB. When, we add angle ABC to the left hand side, we also have to add it to the right hand side. So, we add angle ABC to both the sides. This was one of the most crucial steps in this proof. You will realize that angle ABC is the only angle that is left out and we are looking for 180 degrees. With this step, we get angle ABC plus angle BAC plus angle ABC equal to angle D plus angle ABC. This left hand side as we saw is the sum of all angles of a triangle ACB which equals 180 degrees. The equation we get, is 180 degrees equals angle D plus angle ABC. we’re almost there! Angle ABC is actually angle B of the quadrilateral. So, we can say that angle D plus angle B equal to 180 degrees. So we proved one part. What about the second part? The proof of the second part is very simple. We know that the sum of all angles of a quadrilateral is 360 degrees. Angle A plus angle B plus angle C plus angle D equals 360 degrees. And out of these four, we know that B and D add up to 180 degrees. So, we get angle A plus angle C plus 180 degrees equal to 360 degrees. We just replaced B plus D with 180 degrees. Transposing 180 degrees to the right hand side, we get angle A plus angle C equal to 180 degrees. This was bit long, but remember, the most important part of the proof was to look for equal angles in a segment, and adding an angle to get the sum of all angles of one triangle.