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Home » Quadrangles » Rhombus » Rhombus Diagonals Bisect the Angles

Rhombus Diagonals Bisect the Angles

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A Rhombus has two axes of symmetry, created by the diagonals. Since it is symmetrical, the diagonals bisect the angles, as we will show using triangle congruence.

Problem

In a rhombus ABCD, prove that the diagonals bisect the angles.

i.e prove that ∠BAC ≅ ∠DAC and that ∠BCA ≅ ∠DCA; and also that ∠ADB ≅ ∠CDB and ∠ABD ≅ ∠CBD

Geometry shape: diamond with diagonals

Strategy

Wow, that's a lot of angles and a lot of triangles!

Fortunately, we know so much about the sides, as we are dealing with a rhombus, where all the sides are equal.

We will use triangle congruence to show that the angles are equal, and rely on the Side-Side-Side postulate because we know all the sides of a rhombus are equal.

We will do this first for the angles ∠BAC ≅ ∠DAC and ∠BCA ≅ ∠DCA, following the exact same proof we did for a kite, and then we will repeat the process for the other two pairs of angles, using the other diagonal.

Proof

(1) ABCD is a rhombus //Given

(2) AB=AD                     //definition of rhombus

(3) BC=CD                     //definition of rhombus

(4) AC=AC                    //Common side

(5) △ABC≅△ADC       //Side-Side-Side postulate.     

(6) ∠BAC ≅ ∠DAC      //Corresponding angles in congruent triangles (CPCTC)

(7) ∠BCA ≅ ∠DCA      //Corresponding angles in congruent triangles (CPCTC)

This is just what we did for a kite. Now for the second part:

(1) AD=DC                     //definition of rhombus

(2) AB=BC                     //definition of rhombus

(3) DB=DB                     //Common side

(4) △ADB≅△CDB         //Side-Side-Side postulate.     

(5) ∠ADB ≅ ∠CDB        //Corresponding angles in congruent triangles (CPCTC)

(6) ∠ABD ≅ ∠CBD        //Corresponding angles in congruent triangles (CPCTC)

« Deltoids: One of the Diagonals Bisects the Angles at its Endpoints
Trapezoids: Adjacent angles are Supplementary »

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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