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Properties of Triangle

 

A triangle is a three-edged polygon with three vertices. A fundamental shape in geometry denotes a triangle with vertices A, B, and C. In Euclidean geometry, any three non-collinear points define a unique triangle and, by extension, a unique plane (i.e. a two-dimensional Euclidean space).

A triangle is a three-sided polygon that is sometimes referred to as the trigon. We give the sides special names with a right triangle, with the side opposite the right angle being known as the hypotenuse and the other two sides being known as the legs. We classify triangles into several types: equilateral triangles, right triangles, scalene triangles, obtuse triangles, acute triangles, isosceles triangle, right isosceles, and acute isosceles, obtuse isosceles, acute scalene, right scalene, and obtuse scalene.

Each of the three sides may or may not be equal. This is because each of the three angles may house congruent angles equal to 60° or totally different angles. Therefore, a triangle has 3 sides and is a regular polygon. We can also classify triangles according to their angles. All three angles of an acute triangle are acute (<90°). One right angle and two acute angles make up a right triangle. An obtuse triangle has one acute angle and one obtuse angle (>90°).

Equilateral Triangle

An equilateral triangle is a triangle with the same length on all 3 sides. For equilateral triangles, 3 types of cevians coincide and are equal:

  1. The lengths of the 3 altitudes are the same.
  2. The lengths of the 3 medians are the same.
  3. The lengths of the 3 angle bisectors are equal.

Properties of an equilateral triangle

  1. A triangle is defined by its three sides, three angles, and three vertices.
  2. The sum of a triangle's internal angles is always equal to 180°. This is known as a triangle's angle sum property.
  3. A triangle's exterior angles always add up to 360°.
  4. The sum of consecutive interior and exterior angles lead to 180°.
  5. The length of the remaining side is less than the sum of the lengths of the first two sides of a triangle.
  6. Each of the three sides is equal in an equilateral triangle.
  7. Each of the three angles is congruent and equals 60 degrees.
  8. It has three sides and is a regular polygon.
  9. The perpendicular drawn from the equilateral triangle's vertex to the opposite side divides it into equal halves.

Isosceles Triangle

A triangle with (at least) two equal sides is called an isosceles triangle. The two equal sides in the figure above have the same length, different from the remaining side. This property leads to two triangle angles being equal. As a result, an isosceles triangle has two equal sides and two equal angles.

When two sides are equal, the base angles are also equal, and the perpendicular from the apex angle bisects the base, according to the isosceles triangle property.

Scalene triangle

A scalene triangle has three different angles and no equal length sides. A scalene triangle has the following properties:

  1. It has three sides, each of which is of varying length.
  2. Second, it has three angles, each of a different magnitude.
  3. There is no symmetry at the points.
  4. The triangle's interior angles can be acute, obtuse, or right.
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