Question

State true or false:

Points $M$ and $N$ are taken on the diagonal $AC$ of a parallelogram $ABCD$ such that $AM=CN$, then $BMDN$ is a parallelogram.

• A. True
• B. False

Answer 1

The correct option is A True

Given ABCD is a ${\parallel }^{gm}$ & AM=NC

So AB$\parallel$ DC, AB=DC, AD$\parallel$ BC, AD=BC

$\angle BAC=\angle ACD$ [ Alternate Angles ]

$\angle CAD=\angle BCA$ [ Alternate Angles ]

In $△$ AMB & $△$ CND,

AB=DC

$\angle MAB=\angle NCD$[ Alternate Angles ]

AM=NC

So $△$ AMB$\cong △$ CND

So $\angle CND=\angle AMB,\angle NDC=\angle MBA$

MB=ND ---$\left(1\right)$

Now $\angle DNA=\angle BMC\{{180}^o-\angle CND={180}^o-\angle AMB\Rightarrow \text{Straight Angles}\}$

So MB$\parallel$ ND ---$\left(2\right)$

[ Since $\angle DNA=\angle BMC=$Alternate Angles if MN is transversal for MB & ND ]

Similarly taking $△$ AMD & $△$ CNB, we can prove them to be congruent.

So $\angle CNB=\angle AMD,\angle NBC=\angle MDA$

MD=BN ---$\left(3\right)$

Now $\angle DMC=\angle BNA\{{180}^o-\angle AMD={180}^o-\angle CNB\Rightarrow \text{Straight Angles}\}$

So MD$\parallel$ BN ---$\left(4\right)$[ Since $\angle DMC=\angle BNA=$Alternate Angles if MN is transversal for MD & BN ]

From $\left(1\right),\left(2\right),\left(3\right)\&\left(4\right)$ , we get that BMDN is a parallelogram.