Lesson 78: Classifying Triangles

New Concept

If all three sides are equal in length, the triangle is equilateral. If at least two sides are equal in length, the triangle is isosceles.

What’s next

Next, you’ll apply these definitions by drawing triangles with specific properties and solving problems based on those classifications.

Property

A triangle is equilateral if and only if it is equiangular. This is a biconditional property:

• If a triangle has three equal sides, it must have three equal angles (60° each).
• If a triangle has three equal angles (60° each), it must have three equal sides.

Examples

• Side to Angle: A triangle with side lengths of 8 cm, 8 cm, and 8 cm is equilateral. Because the sides are perfectly balanced, the 180° is split evenly, so every angle is exactly 60°.
• Angle to Side: If a problem states that $\Delta LMN$ has three 60° angles, you immediately know it is equilateral. If side $LM = 3x$ and $MN = 15$, you can solve $3x = 15$ to find $x = 5$.
• Perimeter: If the perimeter of an equilateral triangle is 36 inches, each side must be $36 / 3 = 12$ inches long.

Explanation

If at least two sides are equal in length, the triangle is isosceles. In a triangle, the number of angles with the same measure equals the number of sides with the same measure.

A triangle with side lengths of $7$ cm, $7$ cm, and $4$ cm is isosceles. A right triangle with two equal sides forming the right angle is an isosceles right triangle. If an isosceles triangle has two angles measuring $50^{\circ}$, the third angle is $180^{\circ} - (50^{\circ} + 50^{\circ}) = 80^{\circ}$.

Isosceles might sound fancy, but it just means 'equal legs.' Imagine a triangle with two sides that are twins! These two equal sides are always opposite two equal angles. This type can be tall and pointy or short and wide. It can even have a right angle or an obtuse angle, making it a very versatile shape.

If all three sides have different lengths, the triangle is scalene.

A triangle with side lengths of $3$ cm, $4$ cm, and $5$ cm is a scalene triangle. A scalene triangle can be a right triangle, like one with sides of $6$, $8$, and $10$ units. You can draw a scalene triangle with unique angles of $40^{\circ}$, $60^{\circ}$, and $80^{\circ}$.

The scalene triangle is the rebel of the triangle family, where nothing matches! All three sides have completely different lengths, which means all three angles must have different measures too. You won't find any symmetry here. It’s a unique, one-of-a-kind shape every time, with no two parts being equal, making it perfectly irregular and interesting.