We are asked to prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Let's draw the diagonals and highlight two of the triangles. Let's also label the angles of these highlighted triangles.
Let's summarize what we know about triangles △ DAO and △ BCO.
Congruences
Justification
BO≅DO
AC bisects BD.
AO≅CO
BD bisects AC.
∠ 6≅∠ 5
Vertical angles.
We can see that two sides and the included angle of △ DAO are congruent to the two sides and the inlcuded angle of △ BCO. According to the Side-Angle-Side (SAS) Congruence Postulate, this means that these two triangles are congruent.
△ DAO≅ △ BCO
Corresponding angles of congruent triangles are congruent, so we can compare the other labeled angles.
∠ 3&≅∠ 2
∠ 4&≅∠ 1
Let's use one of these congruent angle pairs.
Since quadrilateral ABCD has two pairs of parallel sides, by definition it is a parallelogram. We can summarize the process above in a two-column proof.
Completed Proof
Given:& ACandDBbisect each other.
Prove:& ABCDis a parallelogram.
Proof:
