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Home » Triangles » Isosceles Triangles » Converse Angle Bisector Theorem for Isosceles Triangles

Converse Angle Bisector Theorem for Isosceles Triangles

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will prove the Converse Angle Bisector Theorem for Isosceles Triangles. We'll show that if a triangle's angle bisector is perpendicular to the opposite side, the triangle is an isosceles triangle.

One of the properties of an isosceles triangle is that the height to the base bisects the apex angle.

The converse of this is also true - If the bisector of an angle in a triangle is perpendicular to the opposite side, the triangle is isosceles.

Problem

In triangle △ABC, the bisector of angle ∠BAC is perpendicular to the base BC. Prove that △ABC is an isosceles triangle.

Properties of Isosceles Triangles

Strategy

As with most converse theorems, we will simply take the proof of the original theory, and use the same strategy (congruent triangles).

We have one common side (AD). We have one pair of congruent angles formed by the angle bisector. AD is the height to the base, which by definition means that the angle it creates with the base (∠ADB) is a right angle, and so is ∠ADC.

So ∠ADB is congruent to ∠ADC and we have our second pair of congruent angles. We can now show that the triangles are congruent. And as a result, AB=AC and the triangle is isosceles.

Proof

(1) ∠BAD ≅ ∠CAD         //Given, AD is the angle bisector
(2) AD=AD                      // Common side, reflexive property of equality
(3) m∠ADB =m∠ADC = 90°   // Definition of height to base
(4) ∠ADB ≅∠ADC //(3), definition of congruent angles
(5) △ABD≅△ACD                  // angle-side-angle
(6) AB=AC        // Corresponding sides in congruent triangles (CPCTC)
(7) ΔABC is isosceles  //(6), definition of isosceles triangle

« Perpendicular Bisector Theorem
Angle Bisector in a Triangle »

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…

Geometry Topics

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