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Home » Lines and Angles » Intersecting Lines and Angles » Angle Bisector Equidistant Theorem

Angle Bisector Equidistant Theorem

Last updated: Apr 29, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

When two rays intersect at a point, they create an angle, and the rays form the two sides of this angle. A line that splits this angle into two equal angles is called the angle bisector.

The Angle Bisector Equidistant Theorem state that any point that is on the angle bisector is an equal distance ("equidistant") from the two sides of the angle.

The converse of this is also true. If a point lies on the interior of an angle and is equidistant from the sides of the angle, then a line from the angle's vertex through the point bisects the angle.

Problem

Angle Bisector with 2 distances

AD is the angle bisector of angle ∠BAC (∠BAD≅ ∠CAD). Show that for any point D, the perpendicular distances |DC| and |DB| are equal.

Strategy

This is a simple proof using congruent triangles - which is the strategy of first choice when we need to show that two things are equal. In this case, to show that the distance between the point on the bisector and the two sides of the angle is equal.

The triangles are already present in the problem's drawing - △ABD and △ACD. They share a common side (AD).

One of their angle pairs is a right angle (as that is the definition of distance) and the other pair of angles is equal since AD is the bisector - and we can show the triangles are congruent using the Angle-Side-Angle postulate.

Proof

(1) AD=AD //Common side, reflexive property of equality
(2) ∠BAD≅ ∠CAD //Given, AD is the angle bisector of ∠BAC
(3) m∠DCA=m∠DBA=90° //definition of distance.
(4) ∠ADC≅ ∠ADB //(2),(3), Sum of angles in a triangle
(5)△ABD≅△ACD //(2),(3),(4), Angle-Side-Angle Postulate
(6) |DB|=|DC| //Corresponding sides of congruent triangles (CPCTC)

« Angle Bisector Theorem
Converse of the Angle Bisector Equidistant Theorem »

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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