Isosceles Triangle
An isosceles triangle has at least two sides of equal length, and the angles opposite those equal sides are also equal.
Formula
\text{base angles are equal when two sides are equal}
Definition
An isosceles triangle has at least two sides the same length (the legs), and the two angles at the base (opposite the equal sides) are also equal, the Isosceles Triangle Theorem; the converse holds too: if two angles of a triangle are equal, the sides opposite them are equal. For legs of length $a$ and base $b$, the base angles each measure $\arccos(b/(2a))$, the apex angle measures $\pi - 2\arccos(b/(2a))$, and the altitude from apex to base is $h = \sqrt{a^2 - b^2/4}$, dividing the triangle into two congruent right triangles.
Example
If a triangle has sides of $5$ cm, $5$ cm, and $3$ cm, it is isosceles, with equal angles at the ends of the $3$ cm base; an A-frame house shape is an isosceles triangle. With legs $=10$ and base $=8$: height $= \sqrt{10^2-4^2}=\sqrt{84}=2\sqrt{21}$, and if the apex angle is $40^\circ$, each base angle is $(180-40)/2=70^\circ$. For legs $a=5$, base $b=6$: $h=\sqrt{25-9}=4$, area $=12$, base angles $=\arccos(3/5)=53.13^\circ$ each.
Key Insight
Isosceles means "equal legs" in Greek, and because of its one line of symmetry, folding an isosceles triangle along that line makes both halves match perfectly; an equilateral triangle is the special case where all three sides are equal, giving three lines of symmetry instead of one. The Isosceles Triangle Theorem is one of Euclid's earliest propositions (Book I, Prop. 5), nicknamed "Pons Asinorum" (Bridge of Asses) because it was historically the first proof in the Elements that students found difficult, and it is the gateway to understanding triangle congruence and symmetry arguments.