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Home » Quadrangles » Squares » The Diagonals of a Square Bisect Each Other

The Diagonals of a Square Bisect Each Other

Last updated: Oct 29, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In a square, the diagonals bisect each other. This is a general property of any parallelogram. And as a square is a special parallelogram, which has all the parallelogram's basic properties, this is true for a square as well.

We have already proven this property for any parallelogram. And today, we will repeat this proof here specifically for square.

Problem

ABCD is a square. Show that its diagonals bisect each other, that is prove that AO=OC and BO=OD.

Square with diagonals


Strategy

When we attempt to prove that the diagonals of a square bisect each other, we will use congruent triangles. This is exactly what we did in the general case, and it's the simplest way to show that two line segments are equal.

In a square, all the sides are equal by definition. So if we look at the triangles formed by the diagonals and the sides of the square, we already have one equal side to use in the Angle-Side-Angles postulate.

The angles are congruent as the sides of the square are parallel, and the angles are alternate interior angles.

Proof

(1) ABCD is a square        //Given
(2) AD = BC                           //(1) , Definition of a square, all sides are equal
(3) AD||BC //(1), a square is a parallelogram, opposite sides are parallel
(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)

« Properties of Rhombus: The Opposite Angles are Congruent
Ratio of Area of Triangles in a Trapezoid »

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…

Geometry Topics

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