Question

In the figure, $DE\parallel OQ$ and $DF\parallel OR$ prove that $EF\parallel QR$

Answer 1

In $△PQO,DE\parallel OQ$

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

$\frac{PE}{EQ}=\frac{PD}{DO}$ $......\left(1\right)$

In $△PRO,DF\parallel OR$

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

$\frac{PF}{FR}=\frac{PD}{DO}$ $......\left(2\right)$

From $eq.\left(1\right)$ and $\left(2\right)$

$\frac{PF}{FR}=\frac{PE}{EQ}$

In $△PQR,\frac{PF}{FR}=\frac{PE}{EQ}$

Line $EF$ divides the $PQ$ and $PR$ of $△PQR$ in the same ratio.

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

$\therefore EF\parallel QR$ [proved]