 Welcome friends to another session on problem solving and this is again related to lines and angles and I'm happy to present one problem here which has got some practical relevance and you see this is a problem in physics actually and you might have studied light and reflection of light so you see how we use the concepts of lines and angles in solving problems related to physics so the questions is something like this there are two plane mirrors pq and rs okay they are placed parallel to each other and there's an incident ray ab which is striking the mirror pq at point b and is it getting reflected as ray bc now ray bc goes and strikes the mirror rs and gets reflected as cd we have to prove that ab is parallel to cd the incident ray is parallel to you know so the incident ab ray is parallel to cd which is the double deflected ray now if you imagine let's say if this portion of the mirror was not there so that in that case what would happen this ray would have gone straight isn't it so if you see you had a torch let's say here in this case you had some torch and you you know switch it on so in hence you can actually shift the beam by this particular mechanism isn't it and the beam beam would be parallel to the initial beam only so hence initially if there were no mirror there was no mirror let's say then the rays would have flown like that or gone like that right but using this mechanism you actually laterally shifted the beam so hence you can displace the beam by this mechanism so this is very very useful in various applications okay now how to prove the given thing so how to go about the proof now if you have learned physics and reflection of light you know that angle of incidence is equal to angle of reflection right this is what we have studied in physics so let's say this angle is i and this angle is r then i is equal to r and again let's say this angle is i one and this angle is r one so i one is also equal to r one now if you see these two lines are perpendicular so let me name it let me know this is this is nothing but the normal right so angle of incidence is equal to angle of reflection and you know we know the nomenclature so this let's say this is bn one and this is cn two so bn one is a normal bn one is normal normal two pq and what is a normal guys normal makes an angle of 90 degree with the surface so this is 90 degrees and similarly so hence we can write bn one is perpendicular to pq similarly you can say cn two is perpendicular to rs so n one and n two are two points don't don't don't get confused so two points named as n one and n two okay so cn two is parallel to or sorry perpendicular to rs so that means bn one is parallel to cn two so this means bn one is parallel to cn two isn't it that means what angle r is equal to i one see if these two lines are parallel then i one must be equal to r right and this is because of alternate interior angles okay so r is equal to i one and we know that from one and two if you see one and two so i is equal to r and i one is equal to r one so hence you can also say if you replace r by i and i one by r one both will be same because r is equal to i and i one is equal to r one so hence this is also true so if you see so these are the two new equations so let's say this equation was three and this was four so add three and four adding three and four we will get r plus i in the left hand side will be equal to i one plus r one in the right hand side now let us understand what is i one and r one if you see here i and r is nothing but i plus r can i write this as angle a bc is equal to angle bcd i one plus r one is bcd so hence angle a bc is equal to bcd right so if i have to you know denote them so this angle is equal to is equal to this angle and if you see this is nothing but a b and cd are two lines and bc is a transversal and they look like they are alternate interior angles so hence you can say a b is parallel to cd why because alternate interior angles are equal okay which two alternate interior angles so basically a bc this one is equal to bcd this one right so hence we could prove that a b is parallel to cd so hence after double reflection the ray is become parallel right so this was the first reflection and this was this was the first reflection and this is the second reflection two reflections are happening and after a second reflection then the ray which is emerging out is parallel to the incident ray that's what we learn and it has got a lot of practical and physical significance in physics