A triangle is one of the basic shapes of 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
ABC .
In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).
Types of triangles
By relative lengths of sides
Triangles can be classified according to the relative lengths of their sides:
- In an equilateral triangle , all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
- In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length.
- In a scalene triangle , all sides and angles are different from one another.
By internal angles
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 in the right triangle. The other two sides are the legs or catheti (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: a 2 + b 2 = c 2 , 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.
- Triangles that do not have an angle that measures 90° are called oblique triangles .
- A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle .
- A triangle that has one angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle .
A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and therefore such triangle is equilateral.
Basic facts
Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a 2-simplex (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements , around 300 BC.
The measures of the interior angles of a triangle in Euclidean space always add up to 180 degrees. This allows determination of the measure of the third angle of any triangle as soon as the measure of two angles are known. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.
The sum of the lengths of any two sides of a triangle always exceeds the length of the third side, a principle known as the triangle inequality . Since the vertices of a triangle are assumed to be non collinear, it is not possible for the sum of the length of two sides be equal to the length of the third side.
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:
- 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.
Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)
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.
- Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those in another right triangle.
- 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.
An important case:
- 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 Hypothenuse-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.
Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.
A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c , and the legs have lengths a and b , then the theorem states that
The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side c .
Some other facts about right triangles:
- The acute angles of a right triangle are complementary.
- If the legs of a right triangle have the same length, then the angles opposite those legs have the same measure. Since these angles are complementary, it follows that each measures 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times √2.
- In a right triangle with acute angles measuring 30 and 60 degrees, the hypotenuse is twice the length of the shorter side, and the longer side is equal to the length of the shorter side times √3 :
For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosine rule and sine rule ).
Points, lines and circles associated with a triangle
There are hundreds of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' t
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