Isosceles triangles and triangle inequalities
Isosceles triangles have a built-in symmetry linking equal sides to equal angles. Inequality theorems tell you which side lengths can form a triangle at all.
Parts of an isosceles triangle
In an isosceles triangle, the two congruent sides are the legs and the third side is the base. The angle between the legs is the vertex angle, and the two angles at the base are the base angles.
Base angles theorem
If two sides of a triangle are congruent, the angles opposite them are congruent: the base angles of an isosceles triangle are equal. The converse also holds: if two angles are congruent, the sides opposite them are congruent. If the vertex angle is 40 degrees, each base angle is (180 - 40)/2 = 70 degrees.
The line of symmetry
The bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base. It splits the triangle into two congruent right triangles, which is how the height of an isosceles triangle is usually found.
Equilateral triangles
Applying the base angles theorem in every direction shows that an equilateral triangle is equiangular, with every angle 60 degrees. Conversely, an equiangular triangle is equilateral.
The triangle inequality
The sum of any two sides of a triangle must be greater than the third side. Lengths 3, 4, and 5 form a triangle, but 2, 3, and 6 do not, since 2 + 3 < 6. Given two sides of 5 and 9, the third side must be between 9 - 5 = 4 and 9 + 5 = 14, exclusive.
Sides and angles
In any triangle, the longest side is opposite the largest angle, and the shortest side opposite the smallest angle. If a triangle's angles are 30, 70, and 80 degrees, the longest side is opposite the 80-degree angle.
The hinge theorem
If two triangles have two pairs of congruent sides, the triangle with the larger included angle has the longer third side, like opening a hinge. This explains why the distance between your feet grows as the angle between your legs widens.
Worked example
An A-frame roof has two equal rafters meeting at a 110-degree peak. Each base angle is (180 - 110)/2 = 35 degrees. If the rafters are each 6 m long, the triangle inequality says the span must be less than 12 m; the line of symmetry splits the roof into two congruent right triangles for calculating its height.
Key takeaways
An isosceles triangle has two congruent legs, a base, a vertex angle, and two congruent base angles; the converse also holds. The vertex angle's bisector is the perpendicular bisector of the base, creating two congruent right triangles. Equilateral triangles are equiangular, with 60-degree angles. Any two sides of a triangle must sum to more than the third, so the third side lies between their difference and their sum. The longest side is opposite the largest angle, and the hinge theorem links a larger included angle to a longer opposite side.
Flashcard
Congruent sides of an isosceles triangle
Check your understanding
Two sides of a triangle are 5 and 9. Which could be the third side?