In the given figure,$l \parallel m$ and $M$ is the mid-point of a line segment $AB$. Show that $M$ is also the mid-point of any line segment $CD,$ having its end points on $l$ and $m,$ respectively.

In $\Delta PQR$,$X$ is the midpoint of $QR$. $XY$ and $XZ$ are altitudes from $X$ on $PQ$ and $PR$ respectively. If $XY = XZ$,then prove that $\Delta PQR$ is an isosceles triangle.

Prove that in a triangle,other than an equilateral triangle,the angle opposite to the longest side is greater than $\frac{2}{3}$ of a right angle.

$ABC$ and $DBC$ are two triangles on the same base $BC$ such that $A$ and $D$ lie on the opposite sides of $BC$,$AB = AC$ and $DB = DC$. Show that $AD$ is the perpendicular bisector of $BC$.

$ABC$ is an isosceles triangle with $AB = AC$ and $D$ is a point on $BC$ such that $AD \perp BC$. To prove that $\angle BAD = \angle CAD$,a student proceeded as follows:
In $\triangle ABD$ and $\triangle ACD$:
$AB = AC$ (Given)
$\angle B = \angle C$ (because $AB = AC$)
and $\angle ADB = \angle ADC$
Therefore,$\triangle ABD \cong \triangle ACD$ $(AAS)$
So,$\angle BAD = \angle CAD$ $(CPCT)$
What is the defect in the above arguments?

$D$ is a point on the side $BC$ of a $\triangle ABC$ such that $AD$ bisects $\angle BAC$. Then