A triangle is a three-sided shape whose three inner angles must sum to 180 degrees. The largest angle will be across from the longest side while the smallest angle will be across from the shortest side of the triangle. If and only if two sides of a triangle are equal, the angles opposite them will be equal as well.
You name triangles by their vertices, so a triangle with vertices A, B, and C is designated as ΔABC.
Properties of Triangles
The sum of three interior angles in any triangle is 180 degrees.
The sum of any two sides of a triangle must be greater than the third side of a triangle.
The longest side of a triangle is opposite the largest angle of a triangle. Conversely, the smallest side of a triangle is opposite the smallest angle of a triangle.
Exterior angle of a triangle is equal to sum of two remote interior angles.
Types of Triangles
You can identify triangle types by the measurements of their sides and angles.
Scalene: A scalene triangle has no equal sides and no equal angles.
Isosceles: An isosceles triangle has at least two equal sides, and the measures of the angles opposite those two sides are also equal to each other.
Equilateral: An equilateral triangle has three sides of equal lengths and three 60-degree angles.
Right Triangle: A right triangle has one angle that measures 90 degrees. The side opposite the right angle is called the hypotenuse.
A triangle with two equal sides is called isosceles. The angles opposite the equal sides are called the base angles, and they are congruent (equal). A triangle with all three sides equal is called equilateral, and each angle is 60 degrees. A triangle with no equal sides (and therefore no equal angles) is called scalene.
Area of a Triangle
Area of a triangle is given by:
A = ½bh
A stands for area, b is the length of the base or bottom of the triangle, and h stands for the height (or altitude), which is the distance that a perpendicular line runs from the base to the angle opposite the base.
Triangles are similar when they have exactly the same angle measures. Similar triangles have the same shape, even though their sides may have different lengths. The corresponding sides of similar triangles are in proportion to each other. The heights of the two triangles are also in proportion.
Two triangles are similar if:
- Corresponding angles are of the same measurement.
- The perimeter of each triangle is in the same ratio as the sides.
- Corresponding sides are all in the same proportion.
To check similarity of triangles, AAA, SAS or SSS rule can be used.