Using ruler and compasses only, construct a parallelogram ABCD us… - Vidyadip

Question

Using ruler and compasses only, construct a parallelogram $\text{ABCD}$ using the following data$: AB = 6 \ cm, AD = 3 \ cm$ and $\angle DAB = 45^o$. If the bisector of $\angle DAB$ meets $DC$ at $P$,prove that $\angle APB$ is a right angle.

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Answer

Steps:
$1$. draw $A B=6 \ cm$.
$2$. With $A$ as a center draw a line $A X$ such that $\angle B A X=45^{\circ}$.
$3$. With $A$ as a center and radii, $3 \ cm$ draw an arc on $AD$.
$4$. now with $D$ and $B$ as a center and radii $6 \ cm$ and $3 \ cm$ draw arcs cutting each other at $C$.
$5$. Join $DC$ and $BC$.
$\text{ABCD}$ is the required parallelogram.
Here
$\angle PAB =\angle APD\dots ...[$ Alternate angles $]$
$\angle C P B=\angle PBA \dots...[$ Alternate angles $]$
Now,
$\angle DPA +\angle APB +\angle CPB =180^{\circ} \ldots \ldots . \text { (i) }$
Also, considering $\text{APB},$
$\angle PAB +\angle PBA +\angle APB =180^{\circ} \ldots \ldots . \text { (ii) }$
Therefore, from $(i)$ and $(ii)$
$\angle APB =90^{\circ}$
Hence proved.

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