Question

State whether the following statement is true or false.

The line segment joining the midpoint of two sides of a triangle is parallel to the third side.

• A. True
• B. False

Answer 1

The correct option is A True

$ABCD$ is a triangle where $E$ and $F$ are mid-points of $AB$ and $AC$ respectively.

Since $AB\parallel CD$ with transversal $ED$

$\angle AEF=\angle CDF$ (alternate angles) .........$\left(1\right)$

In $△AEF$ and $△CDF$

$\angle AEF=\angle CDF$ from $\left(1\right)$

$\angle AFE=\angle CFD$ (vertically opposite angles)

$AF=CF$ as $F$ is the mid-point of $AC$

$\therefore △AEF\cong △CDF$ using AAS rule

So,$EA=DC$ by CPCT

But $EA=EB$ since $E$ is the mid-point of $AB$

Hence,$EB=DC$

Now,in $EBCD,EB\parallel DC$ and $EB=DC$

Thus, one pair of opposite sides are equal and parallel.

Hence,$EBCD$ is a parallelogram.

Since opposite sides of parallelogram are parallel.

So,$ED\parallel BC$

$\therefore EF\parallel BC$

Hence the line segment joining the midpoint of two sides of a triangle is parallel to the third side.