Finding the value of angles inside a triangle

Question

In the following illustration $\angle RPQ = 50^\circ$ and $\angle PSR = 60^\circ$, then side $QS$ is congruent with side $SP$ if:

$(1)$ $\angle RPS + \angle PQR = 50^\circ$

$(2)$ $\angle RPS + \angle QRP = 120^\circ$

According to the answer sheet, only $(1)$ is correct, but I say both are: If I need sides $QS$ and $SP$ to be congruent then $\angle SPQ$ and $\angle PQR$ need both to be $30^\circ$, if $\angle SPQ$ is $30$ then $\angle RPS$ is $20$ so $\angle RPS + \angle PQR = 50^\circ$ so $(1)$ is correct.

If $\angle RPS + \angle QRP = 120$ then that means that $\angle SPQ = 30$, and $\angle QRP = 100$, then $\angle PQR = 30$ because $\angle RPQ = 50 + \angle PQR = 30 + \angle QRP = 100 = 180^\circ$ and $QS=SP$.

What am I doing wrong?

Answer 1

yes, both are correct PS=SQ, means triangle SPQ is isoscale triangle So, we can easily say angle SPQ=angle PQS angle RPS + angle PQR =50 degree

Now for prove second We know, Summation of all inner angle of a triangle is 180 degree Now, angle PSR=60 degree So, summation of other two angles must be 120 degree

Hence both are correct

Answer 2

You are showing "from $PS=SQ$ follows (1) and (2)", whereas you must show backwards. The condition $\measuredangle PRS+\measuredangle RPS=120$ is a necessary but not sufficient condition.