Definition
A triangle is one of the basic shapes in geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .
Triangle properties
>>Vertex:
The vertex (plural: vertices) is a corner of the triangle. Every triangle has three vertices.
>>Base:
The base of a triangle can be any one of the three sides, usually the one drawn at the bottom. You can pick any side you like to be the base. Commonly used as a reference side for calculating the area of the triangle. In an isosceles triangle, the base is usually taken to be the unequal side.
>>Altitude:
The altitude of a triangle is the perpendicular from the base to the opposite vertex. (The base may need to be extended). Since there are three possible bases, there are also three possible altitudes. The three altitudes intersect at a single point, called the orthocenter of the triangle. See Orthocenter of a Triangle.
In the figure above, you can see one possible base and its corresponding altitude displayed.
>>Median:
The median of a triangle is a line from a vertex to the midpoint of the opposite side. The three medians intersect at a single point, called the centroid of the triangle. See Centroid of a Triangle
>>Area
See area of the triangle and Heron's formula
>>Perimeter
The distance around the triangle. The sum of its sides. See Perimeter of a Triangle
>>Interior angles
The three angles on the inside of the triangle at each vertex. See Interior angles of a triangle
>>Exterior angles
The angle between a side of a triangle and the extension of an adjacent side. See Exterior angles of a triangle
Types of triangle
Euler diagram of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.
By relative lengths of sides
Triangles can be classified according to the relative lengths of their sides:
A triangle is one of the basic shapes in geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .
- Area = ½ × b × h
b = base
h = vertical height - Perimeter = a+b+c
Triangle properties
>>Vertex:
The vertex (plural: vertices) is a corner of the triangle. Every triangle has three vertices.
>>Base:
The base of a triangle can be any one of the three sides, usually the one drawn at the bottom. You can pick any side you like to be the base. Commonly used as a reference side for calculating the area of the triangle. In an isosceles triangle, the base is usually taken to be the unequal side.
>>Altitude:
The altitude of a triangle is the perpendicular from the base to the opposite vertex. (The base may need to be extended). Since there are three possible bases, there are also three possible altitudes. The three altitudes intersect at a single point, called the orthocenter of the triangle. See Orthocenter of a Triangle.
In the figure above, you can see one possible base and its corresponding altitude displayed.
>>Median:
The median of a triangle is a line from a vertex to the midpoint of the opposite side. The three medians intersect at a single point, called the centroid of the triangle. See Centroid of a Triangle
>>Area
See area of the triangle and Heron's formula
>>Perimeter
The distance around the triangle. The sum of its sides. See Perimeter of a Triangle
>>Interior angles
The three angles on the inside of the triangle at each vertex. See Interior angles of a triangle
>>Exterior angles
The angle between a side of a triangle and the extension of an adjacent side. See Exterior angles of a triangle
Types of triangle
Euler diagram of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.
By relative lengths of sides
Triangles can be classified according to the relative lengths of their sides:
In an isosceles triangle, two sides are equal in length.[note 1][2] An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.[2] The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 Right Triangle, which appears in the Tetrakis square tiling, is isosceles.
In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
In a scalene triangle, all sides are unequal, and equivalently all angles are unequal. Right triangles are scalene if and only if not isosceles.
By internal angles
Triangles can also be classified according to their internal angles, measured here in degrees.
Triangles can also be classified according to their internal angles, measured here in degrees.
A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle. The other two sides are called the legs or catheti[4] (singular: cathetus) of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, and 5 are a Pythagorean Triple. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.
A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If the greatest side length is c, then a2 + b2 > c2.
- A triangle that has one interior angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle. If the greatest side length is c, then a2 + b2 < c2.
- A "triangle" with an interior angle of 180° (and collinear vertices) is degenerate.
- A right degenerate triangle has collinear vertices, two of which are coincident.
Similarity and congruence Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.
A few basic theorems about similar triangles:
A few basic theorems about similar triangles:
- If two corresponding internal angles of two triangles have the same measure, the triangles are similar.
- If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
- If three corresponding sides of two triangles are in proportion, then the triangles are similar.[note 3]
Some sufficient conditions for a pair of triangles to be congruent are:
- SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure.
- ASA: Two interior angles and the included side in a triangle have the same measure and length, respectively, as those in the other triangle. (The included side for a pair of angles is the side that is common to them.)
- SSS: Each side of a triangle has the same length as a corresponding side of the other triangle.
- AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as AAcorrS and then includes ASA above.)
- Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those in another right triangle. This is also called RHS (right-angle, hypotenuse, side).
- Hypotenuse-Angle Theorem: The hypotenuse and an acute angle in one right triangle have the same length and measure, respectively, as those in the other right triangle. This is just a particular case of the AAS theorem.
- Side-Side-Angle (or Angle-Side-Side) condition: If two sides and a corresponding non-included angle of a triangle have the same length and measure, respectively, as those in another triangle, then this is not sufficient to prove congruence; but if the angle given is opposite to the longer side of the two sides, then the triangles are congruent. The Hypotenuse-Leg Theorem is a particular case of this criterion. The Side-Side-Angle condition does not by itself guarantee that the triangles are congruent because one triangle could be obtuse-angled and the other acute-angled.