question_answer

In \[\Delta DEF\]and\[\Delta PQR,DE=DF,\]\[\angle F=\angle P\] and \[\angle E=\angle Q.\]The two triangles are

A)  Isosceles but not necessarily congruent

B)         Isosceles and congruent

C)         Congruent but not isosceles

D)         Neither congruent nor isosceles

Correct Answer: A

Solution :

In \[\Delta DEF,\,DE=DF.\]So,\[\Delta DEF\]is isosceles. \[\therefore \]\[\angle F=\angle E\]                                     ?(i) Also, \[\angle F=\angle P\]and \[\angle E=\angle Q\]                       ?(ii) From (i) and (ii), we get \[\angle P=\angle Q\]         Now, in \[\Delta PQR,\angle P=\angle Q\Rightarrow RQ=PR\] So, \[\Delta PQR\]is isosceles. Hence, \[\Delta DEF\]and \[\Delta PQR\]are isosceles but not necessarily congruent.