 Hello and welcome to the session. In this session we will discuss some properties of a triangle. We know that a triangle in which two sides are equal is called an isosceles triangle. Consider this triangle ABC. In this we are given that AB is equal to AC. So we can say that triangle ABC is an isosceles triangle. We have an important result which is indeed true for any isosceles triangle that is angles opposite to equal sides of an isosceles triangle are equal. Like as you can see that ABC is an isosceles triangle in which AB is equal to AC. So the angles opposite equal sides that is angle B and angle C are equal. Now another important result is the sides opposite to equal angles of a triangle are equal. Like in triangle ABC we have angle B is equal to angle C. So according to this theorem sides opposite to equal angles are equal. So AB is equal to AC. Consider this figure in which ABC is a right angle triangle at A that is we have angle B AC is given to be 90 degrees and it is also given that AB is equal to AC. Let's find out the measures of the angle B and this angle that is angle AC B. Let's consider this right triangle ABC. So in triangle ABC we have AB is equal to AC which is already given to us and we know that the angles opposite equal sides of an isosceles triangle are equal. So from here we get angle B is equal to angle BCA and we already have angle BAC is equal to 90 degrees since it is a right angle triangle. So by angles and property of triangle we have angle BAC plus angle ACB plus angle B is equal to 180 degrees that is now angle BAC is 90 degrees plus angle ACB or we can also say angle BCA is equal to angle B plus angle B is equal to 180 degrees that is from here we get 2 times angle B is equal to 90 degrees and thus angle B is equal to 45 degrees and we have angle B is equal to angle BCA therefore angle BCA is also equal to 45 degrees. Let's discuss inequalities in a triangle. We have a very important result of inequalities in a triangle which says if two sides of a triangle are unequal the angle opposite to the longer side is larger or greater. Like in this triangle ABC we have BC is greater than AC now the angle opposite the longer side that is angle opposite BC which is angle A is greater than the angle opposite AC which is angle B. Another result is in any triangle the side opposite to the larger or greater angle is longer. Suppose that we are given angle B is greater than angle A so according to this theorem we get that the side opposite angle B which is AC would be greater than the side opposite angle A that is BC. The next important result is the sum of any two sides of a triangle is greater than the third side. Like in this triangle ABC we have AB plus BC is greater than AC. Consider this figure in this figure we are given that angle B is less than angle A angle C is less than angle D and we need to prove that AD is less than BC. Now consider triangle AOB in this we have angle A is greater than angle B and we know that the side opposite the greater angle is longer. So side opposite angle A in triangle AOB that is OB is greater than the side opposite angle B that is OA. Next we consider triangle COD in this we have angle D is greater than angle C. So from here we get that the side opposite angle D which is CO is greater than the side opposite angle C that is DO since the side opposite greater angle is longer. Now on adding these two equations we get CO plus OB is greater than DO plus OA that is CB is greater than DA or we get BC is greater than AD that is we have AD is less than BC. This completes the session hope you have understood the properties of a triangle and inequalities in a triangle.