In the given figure, DE || OQ and DF || OR. Show that EF || QR.

Step 1: Look at the figure and note down the given values
NOTE: DE || OQ, DF || OR.
Show EF || QR
Step 2: According to the given data choose the suitable theorem to simplify
Apply the basic proportional theorem
THEOREM: 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.
EXAMPLE: In ∆POQ, DE || OQ
\frac{PE}{EQ}=\frac{PD}{DO}......................(1)
In ∆POR, DF || OR
\frac{PD}{DO} = \frac{PF}{FR} .................................(2)
Step 3: Compare the equations (1) and (2)
From equations we get \frac{PE}{EQ} = \frac{PF}{FR}
Step 4: Make sure that the sides of required triangle are divided in the same ratio or not.
NOTE: If sides are in the same ratio apply the converse of the
basic proportional theorem.
THEOREM: If a line divides any two sides of a triangle in the same ratio
then the line must parallel to the third side.
EXAMPLE: In ∆PQR,
\frac{PE}{EQ} = \frac{PF}{FR}
So, we can say that EF || QR
Hence, proved EF || QR