State true or false for the following statement:

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, then the chords are equal.

The correct option is A. True

Let $AB$ and $CD$ be two chords intersecting at $E$.

$PQ$ is the diameter passing through $E$.

$AB$ and $CD$ make equal angles with diameter $PQ$.

$\therefore \angle AEQ=\angle DEQ$

Draw $OL\perp AB$ and $OM\perp CD$

In $△OLE$ and $△OME$,

$\angle OLE=\angle OME={90}^{\circ }$

$\angle OEL=\angle OEM$ (Given $\angle AEQ=\angle DEQ$)

$OE=OE$ (common)

$\therefore △OLE\cong △OME$ (By AAS congruence rule)

So, $OL=OM$ (CPCT)

$\Rightarrow$ Chord $AB$ and $CD$ are equidistant from the centre.

Hence, $AB=CD$ (Chords equidistant from the centre are equal in length)

Hence the given statement is true.