 we want to prove that equal quarts of the same circle subtend equal angles at the center. So let me just highlight what is given to us. So this is given to us and what we need to prove is this subtend equal angles at the center. So the diagram on the left shows the situation. We have two equal quarts called PQ and AB. So let's just mark them as congruent and O is the center. So let's highlight the angles POQ and AOB. Now how can we prove this? Again because we see triangles in these two circles we are going to use the method of proving two triangles congruent and because the angle AOB and POQ which we really need to prove that they are equal. So angle POQ needs to be shown to be congruent with angle AOB. These two triangles are part of the triangles that we see and it will just become easier for us to show that if we prove that these two triangles are equal. Now to show the triangle congruent we need at least three corresponding elements to be congruent. So let's consider two triangles in triangle POQ and in triangle AOB. Let's note down what do we observe about different corresponding elements. We know that OP, OQ, OA and OB are all radii right. So let me just mark them with same sign since those will be equal and I can find that OP and OAR corresponding elements. So can I write OA is equal to OP? Yes I can and the reason for it is that both are the radii of the circle. Next corresponding element pair that I can consider is another radii pair. So I had considered OA and OP. So now we can consider OB and OQ. OB is equal to OQ right and these are also radii. Is there any third element that we can show is congruent in these two triangles? Yes we definitely can and that is going to be the given sides AB and PQ and since these are equal quarts AB is equal to PQ and this is given to us right and now we have three corresponding elements from both the triangles to be congruent and therefore triangle POQ is congruent with triangle AOB and since these two triangles are congruent all the other corresponding elements of the both triangles are going to be congruent and therefore this means all the corresponding angles are also congruent and we can say that angle AOB is congruent with angle POQ and this is what we needed to prove and we can conclude that equal quarts of the same circle subtend equal angles at the center. I am using the word same circle but if we have congruent circles and if we have equal quarts in both of them then as well the angles that both of those quarts will subtend at the center will be equal. By the way which property that we used to show these two triangles to be congruent all our sides so SSS that's what we used so we can say that these two triangles are congruent by SSS test of congruency of the angles.