 Hello and welcome to the session that's work out the following problem. It says prove that the degree measure of an arc of a circle is twice the angle subtended by it at any point of the alternate segment of the circle with respect to the arc. In figure 3, o is the centre of the circle, prove that angle x plus angle y is equal to angle z. Now this is the figure 3 and we have to prove that angle x plus angle y is equal to angle z. Let's now move on to the solution and let's first write what is given to us. We are given a circle with centre o angle ACB any point C on the remaining part of the circle what we have to prove that the angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the remaining part of the circle that means we have to prove that angle A o B is twice the angle ACB. And to prove this let's do some construction that we have already done. Join o C and produce it to a point P and join o A and o B that we have already joined. Now we have to join o C and we have to produce it to a point P. Let's now start the proof. Now in triangle A o C o A is equal to o C because these are the radii of the circle angle o C A is equal to angle o AC angle o C A that is this angle is equal to this angle as we know that angles opposite the equal sides are equal angles opposite the equal sides are equal also angle P o A angle P o A that is this angle is equal to angle o C A plus angle o AC at the sum of these two angles because this is the exterior angle of the triangle from above we can see that angle o C A is equal to angle o AC so we have angle P o A is equal to angle o C A plus angle o A o C A because o AC is equal to o C A so this is equal to twice of angle o C A as angle o C A is equal to angle o AC right similarly taking triangle B o C will have angle P o B is twice of angle o C B that is this angle will be twice of angle o C B right let us name this as one and this as two right now adding one and two we have angle P o A plus P o B is equal to twice of o C A plus twice of angle o C B now angle P o A plus P o B is angle a o B if you add them up it is the complete angle a o B and this is twice of angle o C A plus angle o C B now o C A plus o C B is this complete angle that is angle a C B so it is twice of angle a C B so we have proved that angle a o B is twice of angle a C B so this proves the first part of the question now in the second part of the question we have to prove that angle x plus angle y is equal to angle z let's first write what is given to us we are given a circle with center o and we have to prove that angle x plus angle y is equal to angle z now start the proof angle y is the exterior angle of F B that is this triangle angle y is equal to the sum of these two angles that is let's name this as one and two angle one plus angle two so this implies angle two is equal to angle y minus angle one similarly this angle is the exterior angle of the triangle angle a e b let us name this as three so angle three is the exterior angle triangle a e b therefore we have angle three is equal to angle one plus angle x so we have angle three is equal to angle one plus angle x now let us name this as one and this as two also angle z twice the angle two by the above theorem that is angle subtended by an arc at the center is double the angle subtended by it at any point of the circle so angle z is twice the angle two but also angle two is equal to angle three angle subtended in the same segment that is angles in same segment are equal is equal to angle two plus angle two so we have angle z is equal to angle two plus angle three as angle two is equal to angle three so this implies angle z is equal to angle two is angle y minus angle one plus angle three is angle one plus angle x now angle one gets cancelled with angle one and we are left with angle z is equal to angle x plus angle y hence we have proved that angle z is equal to angle x plus angle y so this completes the question while doing the question this is very important to write all the reasons what you are using this is by above theorem so this completes the question and the session bye for now take care have a good day