Question

Suppose that ABCD is a quadrilateral. Suppose that ∠ABM and ∠DCM are right angles and that ∥AB∥=∥DC∥. Prove that the perpendicular bisector of BC is also the perpendicular bisector of AD.

Answer 1

To prove that the perpendicular bisector of BC is also the perpendicular bisector of AD, we show that it passes through the midpoints of both BC and AD and that it is perpendicular to both lines. Given that AB and DC are equal in length and AB is perpendicular to BM and DC is perpendicular to CM, we can conclude that the perpendicular bisector of BC is indeed the perpendicular bisector of AD.

Step-by-step explanation:

In order to prove that the perpendicular bisector of BC is also the perpendicular bisector of AD, we need to show that it passes through the midpoint of both BC and AD, and that it is perpendicular to both BC and AD.

Since ∠ABM and ∠DCM are right angles, we know that AB and DC are perpendicular to BM and CM, respectively. This means that line BM is perpendicular to line AD, and line CM is perpendicular to line BC.

Since ∥AB∥=∥DC∥, we know that AB and DC are equal in length. Therefore, the perpendicular bisector of BC, which passes through the midpoint of BC, also passes through the midpoint of AD. Additionally, since line BM is perpendicular to line AD and line CM is perpendicular to line BC, the perpendicular bisector of BC is also perpendicular to AD.