Question

Write a proof to show that the diagonals of a parallelogram bisect one another. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted.

Answer 1

Statement Reason

ABCD is a parallelogram Given

AD = BC Definition of parallelogram

∠DAO = ∠BCO Alternate interior angles

∠CBO = ∠ADO Alternate interior angles

ΔADO = ΔCBO ASA

BO = DO corresponding parts are congruent

AO = CO corresponding parts are congruent

Step-by-step explanation:

To write the proof, we will use the following parallelogram

By the definition of parallelograms, opposite sides are congruent and parallel, do

AD = BC

Then, ∠DAO = ∠BCO because they are alternate interior angles. They are inside two parallel lines BC and AD and on opposite sides of a transversal CA.

In the same way, ∠CBO = ∠ADO because they are alternate interior angles.

Now, we can say that the triangles ADO and CBO are congruent by ASA (Angle - Side - Angle) and the corresponding part of congruent triangles are congruent so

BO = DO

AO = CO

This means that the diagonal of a parallelogram bisect one another.

Therefore, the proof is

Statement Reason

ABCD is a parallelogram Given

AD = BC Definition of parallelogram

∠DAO = ∠BCO Alternate interior angles

∠CBO = ∠ADO Alternate interior angles

ΔADO = ΔCBO ASA

BO = DO corresponding parts are congruent

AO = CO corresponding parts are congruent