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Home » Quadrangles » Parallelograms » Quadrilateral Whose Diagonals Bisect Each Other

Quadrilateral Whose Diagonals Bisect Each Other

Last updated: Dec 14, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A quadrilateral whose diagonals bisect each other is a parallelogram, as we will show in this exercise. One of the properties of a parallelogram is that its diagonals bisect each other. This is a converse theorem - that shows that if the diagonals bisect each other, the quadrilateral must be a parallelogram.

Problem

ABCD is a quadrilateral with diagonals AC and BD. The diagonals bisect each other: AO = OC and BO = OD. Show that ABCD is a parallelogram.

Parallelogram with diagonals

Strategy

To prove that ABCD is a parallelogram we need to show that both pairs of the opposite sides are parallel to each other. That is the definition of a parallelogram.

To show that lines are parallel, we can use the Converse Alternate Interior Angles Theorem. This converse theorem states that if the interior alternating angles created by the transversal line are congruent, then the two lines intersected by the transversal line are parallel. And to show that angles are congruent, we'll use congruent triangles.

Step 1

The triangles practically present themselves in the above drawing. Let's look at triangles  ΔAOD and ΔCOB. We have two pairs of equal sides: AO = OC and BO = OD. And conveniently, the angles between those two sides (∠AOD , ∠COB) are opposite (vertical) angles, so they are also congruent.

Triangles  ΔAOD and ΔCOB are thus congruent using the Side-Angle-Side postulate. From this, we have ∠OBC≅ ∠ODA as corresponding angles in congruent triangles. ∠OBC and ∠ODA are also alternating interior angles formed by the transversal line segment BD which intersects lines AD and BC. So AD||BC using the Converse Alternate Interior Angles Theorem.

Step 2

Now let's look at another pair of triangles,  ΔAOB and ΔCOD. Again, we know AO = OC and BO = OD. The angles between those two sides (∠AOB, ∠COD) are opposite (vertical) angles, so they are also congruent. Triangles  ΔAOB and ΔCOD are thus congruent using the Side-Angle-Side postulate.

And so, we now have a quadrilateral with two pairs of opposite sides which are parallel. By definition, that is a parallelogram. This proves that the quadrilateral whose diagonals bisect each other is a parallelogram.

From this, we have ∠BAO≅ ∠DCO as corresponding angles in congruent triangles. ∠OBC and ∠ODA are also alternating interior angles, formed by the transversal line segment AC, which intersects lines AB and DC. So AB||DC using the Converse Alternate Interior Angles Theorem.

Proof

(1) AO = OC //Given, ABCD is a quadrilateral whose diagonals bisect each other
(2) BO = OD //Given, ABCD is a quadrilateral whose diagonals bisect each other
(3) ∠AOD≅∠COB // Vertical angles are congruent
(4) ΔAOD ≅ ΔCOB //(1),(2),(3), Side-Angle-Side postulate
(5) ∠OBC≅ ∠ODA //(4) , Corresponding angles in congruent triangles (CPCTC)
(6) AD||BC //(5), Converse Alternate Interior Angles Theorem.
(7) ∠AOB≅∠COD // Vertical angles are congruent
(8) ΔAOB ≅ ΔCOD //(1),(2),(7), Side-Angle-Side postulate
(9) ∠BAO≅ ∠DCO //(8) , Corresponding angles in congruent triangles (CPCTC)
(10) AB||DC //(9), Converse Alternate Interior Angles Theorem.
(11) ABCD is a parallelogram //(6), (10), definition of a parallelogram

« Angle bisector is perpendicular to the base in an isosceles triangle
Area of a Trapezoid with Median »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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