 Hello and welcome to the session. In this session we discussed the following question which says, proof that the ratio of the corresponding sides of two equi-angular triangles is same as the ratio of the corresponding altitudes. First of all, let's recall the AAA similarity criterion, according to this we have that if in two triangles corresponding angles are equal then their corresponding sides are in the same ratio and hence the two triangles are similar. This is the key idea that we use for this question. Now let's proceed with the solution. So consider triangles PQR and P dash Q dash R dash these two triangles are equi-angular. Now we need to prove that the ratio of the corresponding sides of two equi-angular triangles is same as the ratio of the corresponding altitudes. So for triangle PQR, PS is the altitude and for triangle P dash Q dash R dash, P dash S dash is the altitude. So we need to prove that QR upon Q dash R dash is equal to PS upon P dash S dash. We are given that the triangles PQR and P dash Q dash R dash are equi-angular that is the corresponding angles are equal. Angle P is equal to angle P dash, angle Q is equal to angle Q dash, angle R is equal to angle R dash. Now let's once again recall the AAA that is angle angle angle similarity criterion according to this we have that if in two triangles corresponding angles are equal then the corresponding sides are in the same ratio and hence the two triangles are similar. So in triangles PQR and P dash Q dash R dash all the angles that is the corresponding angles are equal and therefore we can say that triangle PQR is similar to the triangle P dash Q dash R dash by angle angle angle similarity criterion and this implies that PQ upon P dash Q dash is equal to QR upon Q dash R dash. We take this as equation 1. Now we consider the triangles PSQ and P dash S dash Q dash in both these triangles we have angle PSQ is equal to angle P dash S dash Q dash equal to 90 degrees as we have PS is the altitude of triangle PQR and P dash S dash is the altitude of triangle P dash Q dash R dash also we have angle Q is equal to angle Q dash as triangle PQR and P dash Q dash R dash are equi-angular so therefore triangle PSQ is similar to the triangle P dash S dash Q dash by AA similarity criterion according to this AA that is angle angle similarity criterion we have that if in two triangles two angles of one triangle are respectively equal to the two angles of the other triangle then the two triangles are similar and so this implies that PS upon P dash S dash is equal to PQ upon P dash Q dash let this be equation 2. Now from equation 1 and equation 2 we have that PS upon P dash S dash is equal to QR upon Q dash R dash we were supposed to prove this so we have proved that in equi-angular triangles the ratio of the corresponding sides is the same as the ratio of the corresponding attitudes so hence proved this completes the session hope we have understood the solution of this question.