In today’s geometry lesson, we will prove that if a dimeter bisects two chords, they are parallel to each other.

## Problem

In circle O, a diameter, AB, bisects two chords, CD and EF. Show that the chords are parallel to each other.

## Strategy

To show two lines are parallel, we can use one of the several methods: either show congruent corresponding angles, congruent alternating interior angles, or the Perpendicular Transversal Theorem.

Since we have shown that the diameter that bisects a chord is perpendicular to that chord. any of these three methods will work, and all are as easy to show. We’ll prove this using each of the three.

## Solution

(1) AB bisects CD // Given

(2) AB⊥CD // A diameter that bisects a chord is perpendicular to that chord

(3) AB bisects EF // Given

(4) AB⊥EF // A diameter that bisects a chord is perpendicular to that chord

(5) ∠OGD ≅∠AHF=90° //(2), (4), definition of perpendicular lines

(6) EF||CD // congruent corresponding angles

alternatively,

(7) ∠OHF ≅∠OGC=90° //(2), (4), definition of perpendicular lines

(8) EF||CD //congruent alternating interior angles

Or,

(9) EF||CD //(2),(4) Perpendicular Transversal Theorem.