Calculadora do teorema do triângulo isósceles

The Isosceles Triangle Theorem (also called the Base Angles Theorem) is one of the oldest theorems in plane geometry — it appears as Proposition 5 of Book I in Euclid's Elements (around 300 BC). It states: if two sides of a triangle are equal, the angles opposite those sides are also equal. Symbolically:

If AB = AC, then ∠B = ∠C.

This tutorial covers the theorem, its converse, the famous "Pons Asinorum" historical proof, and applications in both algebra and proofs.

Defining "isosceles"

An isosceles triangle is one with at least two equal sides. The two equal sides are called the legs and meet at the vertex angle. The third side (often unequal) is the base, and the two angles at its endpoints are the base angles.

Some textbooks define "isosceles" as exactly two equal sides (excluding equilateral). Others use "at least two" (including equilateral as a special case). The inclusive definition is more modern and convenient — every theorem about isosceles triangles also applies to equilateral ones.

The two theorems together

The theorem and its converse together form a powerful "if-and-only-if":

• Direct: If two sides are equal, the opposite angles are equal.
• Converse: If two angles are equal, the opposite sides are equal.

So you can determine isosceles by EITHER condition: see two equal sides OR see two equal angles.

The "Pons Asinorum" — Euclid's famous proof

The proof of the Isosceles Triangle Theorem in Euclid's Elements is known historically as the "Pons Asinorum" ("Bridge of Donkeys") — students who could cross this bridge were considered ready for higher geometry; those who couldn't, were "asses".

The proof: given △ABC with AB = AC, we want to show ∠B = ∠C.

1. Construct the angle bisector from A (call it ray AD with D on BC).
2. AD = AD (reflexive)
3. ∠BAD = ∠CAD (definition of angle bisector)
4. AB = AC (given)
5. △ABD ≅ △ACD by SAS
6. ∠B = ∠C (corresponding parts of congruent triangles — CPCTC)

Modern textbooks typically use this exact 6-step proof. There are alternative proofs (using midpoint, perpendicular foot, etc.) but the angle-bisector approach is the cleanest.

Worked example 1 — find base angles from vertex

An isosceles triangle has vertex angle ∠A = 40°. Find the base angles.

By the theorem, the two base angles are equal. Let each be θ.

40° + θ + θ = 180° (triangle angle sum)
2θ = 140° → θ = 70°.

So ∠B = ∠C = 70°.

Worked example 2 — find vertex from base angles

An isosceles triangle has base angles of 50° each. Find the vertex angle.

vertex = 180° − 2(50°) = 80°.

Worked example 3 — using the converse

In △ABC, ∠B = ∠C = 35°. Prove AB = AC.

By the converse of the Isosceles Triangle Theorem: equal base angles ⇒ equal opposite legs. So AB = AC. QED.

The altitude from the vertex

The altitude from the vertex angle to the base of an isosceles triangle has three special properties (all at the same line):

• It bisects the vertex angle (cuts it into two equal halves).
• It bisects the base (lands at the midpoint of BC).
• It is perpendicular to the base.

This is why an isosceles triangle has a vertical "axis of symmetry" through the vertex angle. The altitude is also the median and the angle bisector — they all coincide. This is unique to isosceles (and equilateral) triangles; in scalene triangles these three lines are distinct.

Area of an isosceles triangle

If the leg length is L and the base length is b, the height from the vertex to the base is:

h = √(L² − (b/2)²)

(by Pythagorean theorem applied to one of the two congruent right triangles formed by the altitude).

Area = ½ × b × h = (b/2) × √(L² − (b/2)²).

Example: L = 5, b = 6. h = √(25 − 9) = 4. Area = 3 × 4 = 12.

The equilateral case

An equilateral triangle is the special case where all three sides are equal. By the isosceles theorem applied to each pair of equal sides, all three angles are equal. By the 180° angle sum: each angle = 60°.

So an equilateral triangle has 3 equal sides AND 3 equal angles AND 3 angles of 60° each. The three properties imply each other.

Recognizing isosceles in problems

Any ONE of these is sufficient to conclude isosceles:

• Two sides are explicitly equal.
• Two angles are explicitly equal.
• The triangle has a line of symmetry.
• An altitude from a vertex also bisects the opposite side.
• An angle bisector from a vertex is also the perpendicular bisector of the opposite side.

Common mistakes

• Confusing isosceles with equilateral. Isosceles = at least two sides equal (or exactly two, depending on definition). Equilateral = all three sides equal. Equilateral is a special case of inclusive isosceles.
• Using the theorem on the wrong angles. The theorem says angles OPPOSITE the equal sides are equal. The vertex angle (between the equal sides) is NOT necessarily equal to anything else.
• Forgetting the converse needs proof too. "Two angles equal ⇒ two sides equal" requires its own proof (or citation of the converse). It is not automatically the same as the direct theorem.
• Treating the altitude formula as for any triangle. The "altitude from vertex bisects base" property is unique to isosceles. In scalene triangles, the altitude lands at a different point than the midpoint.