Problem — Parallel chords and congruent arcs

Text

In a circle with centre $O$, two chords $PQ$ and $RS$ are parallel. Prove that the arcs $PR$ and $QS$ lying between them are congruent.

Solution

Draw the diameter $d$ perpendicular to the two parallel chords: their common perpendicular direction passes through the centre $O$, so such a diameter exists and is unique. Since the perpendicular drawn from the centre to a chord bisects both the chord and the arcs it subtends, $d$ is an axis of symmetry of the whole figure: the reflection in $d$ maps the circle onto itself and swaps $P$ with $Q$ and $R$ with $S$. Consequently the arc $PR$ is mapped onto the arc $QS$; since a reflection is an isometry, the two arcs are congruent. $\blacksquare$