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Home » Quadrangles » Kites (Deltoids) » Deltoids: One of the Diagonals Bisects the Angles at its Endpoints

Deltoids: One of the Diagonals Bisects the Angles at its Endpoints

Last updated: Oct 1, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

Today we will prove one of the properties of deltoids. We will show that one of the diagonals in deltoids bisects the angles at its endpoints.

A deltoid is a quadrilateral with one axis of symmetry. The diagonal that connects the two corners formed by the sides that are equal creates this axis of symmetry.

So we can "fold" the deltoid over this line of symmetry and the two parts will match - meaning that this diagonal also bisects the angles at its endpoints, as we will now prove.

Problem

In a deltoid ABCD, show that the diagonal forming the axis of symmetry bisects the angles at its endpoints.

Geometry shape of a deltoid with diagonal

Show that ∠BAC ≅ ∠DAC and that ∠BCA ≅ ∠DCA

Strategy

As in other similar problems, to show that angles are equal when we have no parallel lines, we will use triangle congruence.

Another hint is that we need to show that the angles are bisected, we know a deltoid is made up of two isosceles triangles, and we have proven that in an isosceles triangle the height to the base bisects the apex angle using a very similar technique.

And in this case, since deltoids are by definition made up of two pairs of equal sides, we are clearly directed to use the Side-Side-Side postulate.

Proof

(1) ABCD is a deltoid    //Given

(2) AB=AD                    //definition of deltoid

(3) BC=CD                    //definition of deltoid

(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)

« Triangle Inequality Theorem
Rhombus Diagonals Bisect the Angles »

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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