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Home » Triangles » Isosceles Triangles » The Height to the Base of an Isosceles Triangle Bisects the Base

The Height to the Base of an Isosceles Triangle Bisects the Base

Last updated: Jan 4, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson we'll learn a simple strategy for proving that in an isosceles triangle, the height to the base bisects the base.

Having proven the Base Angles Theorem for isosceles triangles using triangle congruency, we know that in an isosceles triangle the legs are equal and the base angles are congruent.

With these two facts in hand, it will be easy to show several other properties of isosceles triangles using the same method (triangle congruency).

Let's start by proving that in an isosceles triangle, the height (or altitude)  to the base bisects the base.

Properties of Isosceles Triangles

Problem

Prove that in isosceles triangle ΔABC, the height to the base, AD,  bisects the base.

Strategy

The strategy for this and for the remaining similar problems (showing that the altitude to the base bisects the apex angle, showing that the angle bisector is perpendicular to the base, etc...)  will be the same.

We already have a pair of equal edges (the legs, per the definition of an isosceles triangle) and a pair of congruent angles (per the Base Angles Theorem).  So we will find or construct another pair of congruent angles or another pair of equal sides, and use one of the triangle congruency postulates to show the two triangles are congruent.

So here, we know that AD is the height to the base. By definition, that means that the angle it creates with the base (∠ADB) is a right angle. And, so is ∠ADC,  and we have our second pair of congruent angles.

Proof

(1) ΔABC is isosceles                   // Given

(2) AB=AC                                    // Definition of an isosceles triangle

(3) ∠ACB ≅ ∠ABC                        // Base angles theorem

(4) m∠ADB =m∠ADC = 90°          // Definition of height to base

(5) △ABD≅△ACD                        // angle-angle-side

(6) BD=DC                                    // Corresponding sides in congruent triangles (CPCTC)

« Base Angles Theorem
Isosceles Triangles: the Height to the Base Bisects the Apex Angle »

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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