• Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar
Geometry Help
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
menu icon
go to homepage
search icon
Homepage link
  • About
  • Privacy Policy
  • Contact Me
  • Terms of Service
  • Accessibility Statement
×
Home » Triangles » Isosceles Triangles » Converse Base Angle Theorem

Converse Base Angle Theorem

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will prove that if two angles of a triangle are congruent, the triangle is isosceles. We will use congruent triangles for the proof.

From the definition of an isosceles triangle as one in which two sides are equal, we proved the Base Angles Theorem - the angles between the equal sides and the base are congruent.

Now we'll prove the converse theorem - if two angles in a triangle are congruent, the triangle is isosceles.

Problem

In triangle ΔABC, the angles ∠ACB and ∠ABC are congruent. Prove that ΔABC is isosceles, i.e. that AB=AC.

Base angles theorem in Geometry

Strategy

When proving this theorem, we will do what we usually do with converse theorems. We will try to apply the same strategy we used to prove the original one - the Base Angles Theorem.

There, we drew a line from A to the base BC and proved the resulting triangles are congruent. As a result, the base angles were congruent.

We'll do the same here, prove the triangles are congruent relying on the fact that the base angles are congruent. And as a result, the corresponding sides, AB and AC, will be equal.

And just like in the original theorem, we have a choice of which line to draw. We can draw either the altitude to the base, and use the fact that it creates a linear pair of equal right angles. Or, draw the angle bisector of A, and use the fact that it creates a pair of equal angles at A.

In today's lesson on proving the Converse Base Angle Theorem, we'll provide a proof for both.

Proof

First, we'll draw AD, the height to the base:

Properties of Isosceles Triangles

(1) ∠ACB ≅ ∠ABC       //Given
(2) AD = AD                    // Common side to both triangles, reflexive property of equality
(3) m∠ADC= m∠ADB=90°  //construction
(4) ∠ADC≅∠ADB //(3), definition of congruent angles
(5) △ABD≅△ACD          //(1), (4), (3),  Angle-Angle-Side postulate
(6) AB = AC         // Corresponding sides in congruent triangles (CPCTC)

Now let's repeat it, using AD, the angle bisector:

Base angles theorem with bisected apex

(1) ∠ACB ≅ ∠ABC       //Given
(2) AD = AD                    // Common side to both triangles, reflexive property of equality
(3) ∠CAD≅∠BAD // Given, AD is the bisector
(5) △ABD≅△ACD          //(1), (3), (2),  Angle-Angle-Side postulate
(6) AB = AC         // Corresponding sides in congruent triangles (CPCTC)

« Angle Bisector in a Triangle
Converse of the Scalene Triangle Inequality »

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].

Primary Sidebar

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…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
    • Kites (Deltoids)
    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

By accessing or using this website, you agree to abide by the Terms of Service and Privacy Policy.


Copyright © 2025