Question

What are the steps for Given: start overline, A, D, end overline, \parallel, start overline, B, C, end overline
AD
∥
BC
and start overline, A, B, end overline, \parallel, start overline, C, D, end overline, .
AB
∥
CD
.

Prove: start overline, A, C, end overline
AC
bisects start overline, B, D, end overline
BD
.

Answer 1

Let AC and BD meet at O. We will prove OB = OD, so AC bisects BD.

1. Consider triangles AOB and COD.
- ∠AOB = ∠COD (vertical angles).
- ∠ABO = ∠CDO because AB ∥ CD (alternate interior angles).
Thus ΔAOB ~ ΔCOD (AA), so
AO/CO = BO/DO. (1)

2. Consider triangles AOD and COB.
- ∠AOD = ∠COB (vertical angles).
- ∠ADO = ∠CBO because AD ∥ BC (alternate interior angles).
Thus ΔAOD ~ ΔCOB (AA), so
AO/CO = DO/BO. (2)

3. From (1) and (2) we have BO/DO = DO/BO. Hence (BO)^2 = (DO)^2, and since lengths are nonnegative, BO = DO.

Therefore O is the midpoint of BD and AC bisects BD. QED.