Definition 1: Isoceles Triangle
An isoceles triangle that has two sides of equal length.
Theorem 1: In an isoceles triangle, the base angles are congruent.
(1) Let triangle ABC be an isoceles triangle with AB ≅ AC
(2) Let D be a point that extends AB such that A,B,D are on the same line.
(3) Let E be a point that extends AC such that A,C,E are on the same line and AE ≅ AD.
(4) triangle DAC ≅ triangle EAB by Side-Angle-Side (see here) since:
(a) AB ≅ AC (step #1)
(b) ∠ DAC is a common angle.
(c) AE ≅ AD.
(5) We also know that triangle DBC ≅ triangle ECB by Side-Side-Side (see here if needed) since:
(a) BD ≅ CE since AD ≅ AG (step #3) and AB ≅ AC (step #1)
(b) BE ≅ CD (step #4) [By properties of congruent triangles, see here if needed]
(c) BC is a common side.
(6) So, now it follows that ∠ ABC ≅ ∠ ACB since:
(a) ∠ ABE ≅ ∠ ACD [By Properties of congruent triangles, see here if needed]
(b) ∠ EBC ≅ ∠ DCB [By Properties of congruent triangles and step #5]
(c) And finally, we know that:
∠ ABC = ∠ ABE - ∠ EBC
∠ ACB = ∠ ACD - ∠ DCB
Corollary: If base angles are congruent, then sides are congruent
(1) Let ABC be a triangle such that ∠ ABC ≅ ∠ ACB
(2) Assume that AB does not equal AC.
(3) Then one of them is greater. Let's assume AB. (Otherwise, we can make the same argument for side AC)
(4) Then there exists a point D such that BD is less than AB and BE ≅ AC
(5) From Theorem 1 above, we know that ∠ ABC ≅ ∠ DCB
(6) But this implies ∠ DCB ≅ ∠ ACB which is impossible.
(7) So we have a contradiction and we reject our assumption in #2.
Theorem 2: The angle bisector of an isoceles triangle, is perpendicular to the base and divides up the base into two congruent segments.
(1) Let AD be the angle bisector of ∠ BAC
(2) From this we, see that triangle BAD ≅ triangle CAD by S-A-S (see here) since:
(a) ∠BAD ≅ ∠ CAD since AD is the angle bisector.
(b) AB ≅ AC since ABC is an isoceles triangle.
(c) AD is a shared side between the two triangles.
(3) From congruent triangles (see here), we know that:
BD ≅ DC
∠ ADB ≅ ∠ ADC
(4) Now, since ∠ ADB, ∠ ADC are congruent and add up to 180 degrees (see here), we can conclude that they are both right angles.
- David Joyce, Euclid's Elements, Book I, Proposition 5