question_answer

\[\angle PQR=\angle XYZ.\] If \[\overrightarrow{\text{QM}}\], bisects \[\angle PQR,\] \[\overrightarrow{YN}\] bisects \[\angle XYZ,\] which of the following statements are true?

(i) \[\angle PQM+\angle NYZ=\angle PQR\]

(ii) \[\angle MQR+XYN=\angle XYZ\]

(iii) \[\angle PQM=2\angle PQR\]

(iv) \[\angle XYZ=2\angle MQR\]

A) (i) and (ii) only

B) (i) and (iv) only

C) (ii) and (iii) only

D) (i), (ii) and (iv) only  

Correct Answer: D

Solution :

Given \[\angle PQR=\angle XYZ,\,\,\overrightarrow{QM}\] bisects\[\angle PQR\] and \[\overrightarrow{YN}\] bisects \[\angle \,\,XYZ\] respectively. \[\Rightarrow \angle PQM+\angle MQR=\angle XYN=\angle NYZ\] \[\Rightarrow \angle PQM+\angle MQR=\angle PQR\] is true. \[\angle MQR+\angle XYN=\angle XYZ\] is true \[\angle PQM=2\angle PQR\] is false as \[\angle PQM=\frac{1}{2}\angle PQR.\] \[\angle XYZ=2\angle MQR\] is true since \[2\angle MQR=\angle PQR=\angle XYZ\]. Hence (i), (ii) and (iv) are true.