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Home » Quadrangles » Parallelograms » The Diagonals of a Parallelogram Bisect Each Other

The Diagonals of a Parallelogram Bisect Each Other

Last updated: Jan 4, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In this lesson, we will prove that in a parallelogram, each diagonal bisects the other diagonal.

A line that intersects another line segment and separates it into two equal parts is called a bisector.

In a quadrangle, the line connecting two opposite corners is called a diagonal. We will show that in a parallelogram, each diagonal bisects the other diagonal.

Problem

ABCD is a parallelogram, and AC and BD are its two diagonals.  Show that AO = OC and that BO = OD

Parallelogram with diagonals

Strategy

Once again, since we are trying to show line segments are equal, we will use congruent triangles. And here, the triangles practically present themselves. Let's start with showing that AO is equal in length to OC, by using the two triangles in which AO and OC are sides: ΔAOD and  ΔCOB.

There are all sorts of equal angles here that we can use. Several pairs of (equal) vertical angles, and several pairs of alternating angles created by a transversal line intersecting two parallel lines. So finding equal angles is not a problem. But we need at least one side, in addition to the angles, to show congruency.

As we have already proven, the opposite sides of a parallelogram are equal in size, giving us our needed side.

Once we show that ΔAOD and  ΔCOB are congruent, we will have the proof needed, not just for AO=OC, but for both diagonals, since BO and OD are also corresponding sides of these same congruent triangles.

Proof

(1) ABCD is a parallelogram    //Given
(2) AD || BC                                 //From the definition of a parallelogram
(3) AD = BC                                 //Opposite sides of a parallelogram are equal in size
(4) ∠OBC ≅ ∠ODA                      //Alternate Interior Angles Theorem
(5) ∠OCB ≅ ∠OAD                      //Alternate Interior Angles Theorem
(6) ΔOBC ≅ ΔODA                      // Angle-Side-Angle
(7) BO=OD                                // Corresponding sides in congruent triangles (CPCTC)
(8) AO=OC                                // Corresponding sides in congruent triangles (CPCTC)

The converse of this theorem is also true - if the diagonals of a quadrilateral bisect each other, then that quadrilateral is a parallelogram.

« Isosceles Triangles: the Median to the Base is Perpendicular to the Base
The Diagonals of Squares are Perpendicular to Each Other »

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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About

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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