 Hello and welcome to the session. In this session we will use the properties of similarity of transformations to establish the angle-angle criterion for two triangles to be similar. Now we know similar figures are those figures which have same shape but different size. Now, there is only one transformation which presents shape but does not present size and that transformation does dilution. Thus, we will use dilution to prove the angle-angle criterion of similarity of two triangles. Now let us draw triangle ABC with any measure of angle B at 60 degrees, measure of angle C at 45 degrees, also the angle is equal to 4 centimeters, BC is equal to 3 centimeters and CA is equal to 5 centimeters. So, this is triangle ABC. Now we will draw another triangle PQR which is larger than the triangle ABC such that measure of angle Q is equal to measure of angle B and measure of angle R is equal to measure of angle C. So, we have drawn larger triangle PQR taking PQ is equal to 8 centimeters, QR is equal to 6 centimeters and RP is equal to 10 centimeters and here angle Q is equal to angle B that is 60 degrees and angle R is equal to angle AC that is 45 degrees. Now inside triangle PQR, draw a line segment Q dash R dash parallel to QR such that Q dash R dash is equal to BC is equal to 3 centimeters. So, here we have drawn line segment Q dash R dash parallel to line segment QR and length of this line segment is 3 centimeters. Now we will draw measure of angle Q dash is equal to measure of angle Q and measure of angle R dash is equal to measure of angle R thus angle Q dash is equal to 60 degrees and angle R dash is equal to 45 degrees and we have extended the two rays till they meet each other. Now we label this point of intersection as P dash. Now we see that we have another triangle P dash Q dash R dash such that measure of angle Q dash is equal to 45 degrees. Now we join the corresponding vertices of the two triangles that is triangle PQR and triangle P dash Q dash R dash using line segments. So, we have joined the corresponding vertices of the two triangles using line segments and then we extend these line segments. Now when we extend these line segments we see that these line segments that is these three line segments intersect at one point. Let us label this point as A. So, we have three triangle angles triangle ABC triangle PQR and triangle P dash Q dash R dash. Now let us find measure of angle P. Now we know that in a triangle some of our angles is 180 degrees. So, in triangle PQR angle P plus angle Q plus angle R is equal to 180 degrees which implies angle P plus angle Q is 60 degrees plus angle R is 45 degrees is equal to 180 degrees. This further implies angle P plus 105 degrees is equal to 180 degrees which implies angle P is equal to 180 degrees minus 105 degrees which is equal to 75 degrees. So, angle P is 75 degrees. Similarly, we can find measure of angle P dash. Now in triangle P dash Q dash R dash angle P dash plus angle Q dash plus angle R dash is equal to 180 degrees which implies angle P dash plus angle Q dash plus angle R dash is equal to 180 degrees. This implies angle P dash plus 60 degrees plus 45 degrees is equal to 180 degrees which implies angle P dash plus 105 degrees is equal to 180 degrees. This implies angle P dash is equal to 180 degrees minus 105 degrees which is 75 degrees. So measure of angle P is equal to measure of angle P dash also by construction measure of angle Q is equal to measure of angle Q dash and measure of angle R is equal to measure of angle R dash so corresponding angles are equal. Also we have measure of angle A as 75 degrees now when we measure the size of triangle P dash Q dash R dash we see that P dash Q dash is equal to A G is equal to 4 centimeters P dash is equal to C A is equal to 5 centimeters. This is because the corresponding angles of the two triangles are equal. Now let us see the ratio of corresponding sides of triangle P Q R and triangle P dash Q dash R dash. Now we see that ratio P dash Q dash upon P Q is equal to 4 centimeters which is equal to 1 upon 2 also ratio Q dash R dash upon Q R is equal to 3 upon 6 that is again 1 upon 2 then ratio. R dash P dash upon R P is equal to 5 upon 10 which is also 1 upon 2 so we see that ratio of corresponding sides is also same thus corresponding sides are proportional. From this construction we observe that triangle P dash Q dash R dash is a dilated image of triangle P Q R which center O such that the corresponding angles of two triangles are equal and the corresponding sides of two triangles are proportional with skin factor 1 upon 2 thus triangle P Q R is similar to triangle P dash Q dash R dash thus from this activity we see that we have taken only two corresponding angles equal that is ratio of angle Q dash is equal to measure of angle Q and measure of angle R dash is equal to measure of angle R and we get a similar triangle P dash Q dash R dash thus we conclude that when two corresponding angles are equal for two different triangles then the two triangles are similar. So in this fashion we have learnt to establish angle angle criterion for two similar triangle using transformations and let's complete the fashion hope you all have enjoyed the fashion.