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Home » Triangles » Isosceles Triangles » Isosceles Triangles: the Height to the Base Bisects the Apex Angle

Isosceles Triangles: the Height to the Base Bisects the Apex Angle

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

We can use the proof that in isosceles triangles the height to the base bisects the base, and get a "two-for-one," as the exact same proof shows us multiple corresponding parts that are congruent.

Properties of Isosceles Triangles

Problem

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

Strategy

We will simply reuse what we have done before.

We know that 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 found our second pair of congruent angles. We can now show that the triangles are congruent, and as a result that the angles ∠CAD and ∠BAD are congruent.

Nothing is changed in the proof steps at all!

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) ∠CAD ≅ ∠BAD         // Corresponding angles in congruent triangles (CPCTC)

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

« The Height to the Base of an Isosceles Triangle Bisects the Base
Congruent Right Triangles »

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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