PQR = XYZ . If QM bisects PQR , YN bisects XYZ , which of the following statements are true? I. PQM + NYZ = PQR II. MQR +XYN= XYZ III. PQM =2 PQR IV. XYZ =2 MQR

PQR = XYZ . If QM bisects PQR , YN bisects XYZ , which of the fol…

MCQ

$\angle{\text{PQR}}=\angle{\text{XYZ}}.$ If $\overrightarrow{\text{QM}}$ bisects $\angle{\text{PQR}},$ $\overrightarrow{\text{YN}}$ bisects $\angle{\text{XYZ}},$ which of the following statements are true?
$I.\ \angle{\text{PQM}}+\angle{\text{NYZ}}=\angle{\text{PQR}}$
$II.\ \angle{\text{MQR}}+\text{XYN}=\angle{\text{XYZ}}$
$III.\ \angle{\text{PQM}}=2\angle{\text{PQR}}$
$IV.\ \angle{\text{XYZ}}=2\angle{\text{MQR}}$

A
$(i)$ and $(ii)$ only
B
$(i)$ and $(iv)$ only
C
$(ii)$ and $(iii)$ only
✓
$(i), (ii)$ and $(iv)$ only

✓

Answer

Correct option: D.

$(i), (ii)$ and $(iv)$ only

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