Triangle, in plane geometry, is an enclosed figure that has three line segments for sides. The sides meet at three points called vertices, and each vertex forms an angle with two of the sides. The sum of the three angles of a triangle is always 180°. A triangle is a type of polygon (see Polygon ).
Kinds of triangles.
Triangles can be classified according to the relationships of their sides. A scalene triangle is a figure with three unequal sides. An isosceles triangle has at least two equal sides. A triangle with all three sides of equal length is called an equilateral triangle. Therefore, every equilateral triangle is also an isosceles triangle, but not every isosceles triangle is an equilateral triangle.
Triangles are also classified by their angles. A triangle in which every angle is smaller than 90° is an acute triangle. An obtuse triangle has one angle larger than 90°. A right triangle has one right–that is, 90°–angle. No triangle can have more than one obtuse angle or one right angle.
Properties and relationships of triangles.
The parts of a triangle have many interesting characteristics and relationships. Sometimes a useful relationship between two or more triangles can also be established. Some of the most notable of these properties and relationships are discussed below.
Perimeter and area.
The perimeter of a triangle is the sum of the lengths of its sides. To find the area of a triangle, we must know the altitude, or height, which is the perpendicular distance from a vertex to the opposite side, or base. The area is calculated by multiplying the base by the altitude and then dividing by 2. Even if the perimeters of several triangles are the same, the areas of those triangles may differ.
Right triangles and the Pythagorean theorem.
The sides of a right triangle have a special relationship to each other. This relationship is expressed in a mathematical statement called the Pythagorean theorem (see Pythagorean theorem ). This theorem was known in ancient civilizations, but it is credited to Pythagoras, a Greek philosopher and mathematician. According to the Pythagorean theorem, a triangle is a right triangle if and only if the sum of the squares of the two shorter sides equals the square of the longest side, called the hypotenuse. This statement can be written as a formula: a2 + b2 = c2. For example, if the sides of a triangle are 6, 8, and 10, it is a right triangle because 62 + 82 = 36 + 64 = 100 = 102. Using the formula, we can find the length of any side of a right triangle if we know the lengths of the other two sides. We can also determine whether a triangle has a right angle if we know the lengths of all its sides.
Congruence.
Two triangles are congruent—that is, they have exactly the same size and shape—if all their corresponding sides and angles are equal. Mathematicians have formulated specific sets of conditions for determining congruence. For example, two triangles are congruent if the sides of one are congruent to the corresponding sides of the other. Using S to represent side, this statement can be abbreviated as SSS ≅ SSS. Other sets of conditions for congruency involve angles as well as sides, with A standing for angle. These sets are AAS ≅ AAS, SAS ≅ SAS, and ASA ≅ ASA.
Similarity.
Triangles that are similar have equal corresponding angles and proportional corresponding sides. Similar triangles have the same shape but not necessarily the same size. For example, if the sides of one triangle are 3, 4, and 5, and those of another are 6, 8, and 10, the sides are proportional, and the triangles are similar.
The similarity of two triangles, like the congruence of two triangles, can be established without knowing all the sides and angles. If two sets of corresponding angles are equal, for example, the triangles are similar. If all three corresponding sides of the triangles are proportional, the triangles are also similar. In a right triangle, the altitude from the right angle divides the triangle into two triangles that are similar to each other and to the parent triangle.
See also Geometry (Congruence) (Similarity) ; Trigonometry .
