 another session on congruent triangles so in the last session we have seen what is meant by congruent triangles now we are going to take up the criteria one by one as to what is required for two triangles to be congruent so the criteria if I have to mention let us let me mention the criteria for congruence so one was corresponding corresponding corresponding angles corresponding angles must be equal correct that was point number one must be equal so this is for this holds for any geometric figure not necessarily for triangles right any linear geometric figure and second point was corresponding corresponding corresponding sides are equal sides are equal okay now in this case if you see you will be able to prove that two triangles are congruent let me draw a triangle first so here is what I am going to draw up triangle so triangle ABC okay so this triangle ABC will be congruent too so let me just select all copy and paste right so here is the triangle right so we can prove that what can we prove we can say that triangle ABC ABC is congruent to the symbol for congruence triangle a1 b1 c1 by SAS criteria why so AB is given to be equal to AB is equal to a1 b1 okay secondly let's say BAC angle BAC is equal to angle B1 a1 c1 let's say these conditions are given and what else angle AC or sorry side AC not angle so I have to write side AC is equal to a1 c1 okay so these are the three the three conditions given now we have to prove that so if you see we have to just prove that this is sufficient enough so these conditions are sufficient conditions so these conditions are sufficient that's what they are saying so if these three are met that means the other three sides and angles will also be equal how do we know that so clearly since AB is equal to a1 b1 so if you try and superimpose AB on a1 b1 or a1 b1 on AB it will fit in and since angle a is also equal to angle a1 so this will also fit in okay so and so a1 c1 is also again with that logic a1 c1 will also be fitting in correct so hence there will be a perfect superimposition right so a1 b1 will fit on AB because same length and same point right and angle a is equal to a1 so the two arms of angle a and a1 must coincide so they are coinciding and the third point is AC is given equal to a1 c1 and a1 if coincides with a then automatically since AC is equal to a1 c1 the other end point of these two line segments that is c1 and c will be coincident so that means a1 is coinciding on a b1 is coinciding on b and c1 is coinciding on c so the three vertices are coincident on each other the moment that happens the two triangles become congruent isn't it so hence we can say that this is called side angle side criteria what does it mean if if one side or not two sides so let me write this if two sides two sides and the included angle included now this is very important included angle two sides and included angle of a triangle triangle r equal to r equal to r equal to the corresponding sides two words are included and corresponding very important corresponding corresponding sides and included angle sorry the word is included angle included angle right so if two sides of the and two sides and the included angle of a triangle are equal to the corresponding sides and included angle of another triangle of another triangle the triangles are equal the triangles are congruent triangles are congruent but so this is the first criteria guys sas criteria this criteria is called sas you have to mention this criteria whenever we are going to prove two congruence using these this criteria sas criteria so once again i will lay emphasis on the included part of it see what is meant by included part so included angle so a and a1 a is included angle of ab and ac but b is not included angle of ab and ac b is included angle of ab and bc correct so this sas criteria means that the side then the included angle and the another arm so basically you are just trying to prove one angle is equal to another angle with both the arms also being equal then the two triangles are congruent because they will be super imposing each and super posing each other right so you understood so a is equal to a1 and the arms which are making that angle a so ab and ac are the sides of which the included angle is a so please be very very careful that if you are two sides and let's say angle b was given to be equal then the triangles need not be congruent guys right what i'm saying is if this small c is equal to c1 this b is equal to b1 and let's say this b is equal to b1 right which is not the included angle then the two triangles need not be congruent so it's very very important to understand that this criteria is only for included angle right when that happens then the other parts of the triangle automatically become become equal right so hence two sides and included angle two sides and included angle are equal then the two triangles are congruent