Isosceles Triangle
GeometryAn 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 two sides that are the same length. The two angles at the base (the ends of the unequal side) are also equal.
Example
If a triangle has sides of $5$ cm, $5$ cm, and $3$ cm, it is isosceles. The two angles at the ends of the $3$ cm base are equal. An A-frame house shape is an isosceles triangle.
Key Insight
Isosceles means "equal legs" in Greek. The two equal sides are the "legs" and the unequal side is the "base." Because of its one line of symmetry, if you fold an isosceles triangle along that line, both halves match perfectly.
Definition
An isosceles triangle has at least two congruent sides (legs). The Isosceles Triangle Theorem states: the base angles (angles opposite the equal sides) are congruent. The converse is also true: if two angles of a triangle are equal, the sides opposite them are equal.
Example
Isosceles triangle with legs $= 10$ and base $= 8$: base angles are equal. If vertex angle (top) $= 40^\circ$, each base angle $= (180 - 40)/2 = 70^\circ$. Height to base $= \sqrt{10^2 - 4^2} = \sqrt{84} = 2\sqrt{21}$.
Key Insight
An equilateral triangle is a special case of isosceles (all three sides equal). Every isosceles triangle has exactly one line of symmetry (the perpendicular bisector of the base), while equilateral triangles have three. Isosceles triangles appear in architecture as gable rooflines and in art as natural balanced forms.
Definition
An isosceles triangle has two equal sides of length $a$ and a base of length $b$. The base angles each measure $\arccos(b/(2a))$ and the apex angle measures $\pi - 2\arccos(b/(2a))$. The altitude from apex to base is $h = \sqrt{a^2 - b^2/4}$, dividing the triangle into two congruent right triangles.
Example
For legs $a = 5$, base $b = 6$: $h = \sqrt{25 - 9} = 4$. Area $= (1/2)(6)(4) = 12$. Base angles $= \arccos(3/5) = 53.13^\circ$ each. Apex angle $= 180 - 2(53.13) = 73.74^\circ$.
Key Insight
The isosceles triangle theorem is one of Euclid's earliest propositions (Book I, Prop. 5), nicknamed "Pons Asinorum" (Bridge of Asses) because it was the first proof in the Elements that students historically found difficult. It is the gateway to understanding triangle congruence and symmetry arguments.