 Hello friends, welcome to another problem-solving session on triangles and its angles in the given question PS is the bisector of QPR. So what is PS and QPR so you can see this line PS is a bisector of QPR the whole vertex angle QPR, right and PT is perpendicular to QR We have to show that angle TPS lets Mention the name so TPS is let's say this angle Which angle my friends this angle? Let us call it X Okay is half Q minus R So how to approach this problem? So let's start with writing whatever is given. So given is Angle QPS QPS is equal to angle RPS Is it RPS why because PS is the bisector PS is the bisector Is it Very good. So what next? also PT is perpendicular to QR that means what? Angle PTQ Is equal to angle PTR Correct this is given okay, and we have to prove that to show What is to be shown? Angle TPS is half Angle Q minus angle R So how do we approach such problems? So One angle is given 90 degrees bisector is given One extra information which we know is angle some property. So can we use that? To find the relationship. So TPS Isn't it so you can see there are two right angle triangles And we are going to use both of them, right? So let's say in solution Okay in triangle PQT PQT right angle PQT or I'm simple right angle Q plus angle QPT QPT plus 90 degrees Which is angle T is equal to 180 degrees angle some property angle some property of a Triangle folks, you already know that so what do I get angle Q? Plus angle QPT Is equal to 90 degrees Now if you look at QPT, what is QPT? If you look at very carefully, what is QPT? So and this is angle Q plus QPT if you see QPT This is the angle I'm talking about isn't it this whole angle minus X Is it it whole angle is QPS? So angle QPS minus X Which is our TPS correct This is equal to 90 degrees No problem, right? Now dear friends QPS is equal to Right, so can I not or yeah, so can I not say angle Q plus QPS? Can I not replace it by RPS? Right What is RPS? See QPS is equal to RPS from the very given fact Correct. So this is true. Now. Let me write it here So angle Q plus RPS. Now RPS RPS can be written as see I can write that as angle Q the first one then RPS and And RPS can be written as angle RPT minus X, isn't it? This is RPS RPS is you can check This is RPS this angle. So let me draw with this on this color RPS This is RPS is equal to the full angle which is the full angle this one and minus this X I hope you got it Right So angle Q plus RPT minus X That is RPS and then I had another X. So write another X like that is equal to 90 degrees Isn't it so this implies Q plus angle RPT RPT Right Minus 2x is 90 degrees What is RPT? If you see RPT is nothing but let me write it in a separate note so that it becomes easier So let us this is one now in triangle PTR It's another right angle triangle in this angle RPT RPT plus angle R. I'm simply writing angle R Plus 90 degrees is equal to 180 degrees again ASP Check you'll get that. So angle RPT Right can be written as If you see 180 minus 90 that is 90 degrees minus angle R from this this is 2 now from one and two I Can substitute RPT by to whatever we have got so we can write angle Q Plus RPT I'm removing and replacing it by this 90 minus R So 90 degrees minus angle R Minus 2x is 90 degrees Correct from one and two so this 90 this 90 will go and we'll get 2x is equal to angle Q Minus angle R and hence X is half Angle Q minus angle R and what was X guys TPS? So angle TPS look at the diagram. We'll get it TPS is equal to half Angle Q minus angle R Isn't it? This is what we had to prove and we have proved it you can write hence Proved so what is the learning in this question? Use of angles and properties so whenever some angles are given in the triangle you can always use angles and property to find Other angles of the triangle