<P> In several geometries, a triangle has three vertices and three sides, where three angles of a triangle are formed at each vertex by a pair of adjacent sides . In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, π radians, two right angles, or a half - turn . </P> <P> It was unknown for a long time whether other geometries exist, where this sum is different . The influence of this problem on mathematics was particularly strong during the 19th century . Ultimately, the answer was proven to be positive: in other spaces (geometries) this sum can be greater or lesser, but it then must depend on the triangle . Its difference from 180 ° is a case of angular defect and serves as an important distinction for geometric systems . </P> <P> In Euclidean geometry, the triangle postulate states that the sum of the angles of a triangle is two right angles . This postulate is equivalent to the parallel postulate . In the presence of the other axioms of Euclidean geometry, the following statements are equivalent: </P> <Ul> <Li> Triangle postulate: The sum of the angles of a triangle is two right angles . </Li> <Li> Playfair's axiom: Given a straight line and a point not on the line, exactly one straight line may be drawn through the point parallel to the given line . </Li> <Li> Proclus' axiom: If a line intersects one of two parallel lines, it must intersect the other also . </Li> <Li> Equidistance postulate: Parallel lines are everywhere equidistant (i.e. the distance from each point on one line to the other line is always the same .) </Li> <Li> Triangle area property: The area of a triangle can be as large as we please . </Li> <Li> Three points property: Three points either lie on a line or lie on a circle . </Li> <Li> Pythagoras' theorem: In a right - angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides . </Li> </Ul>

Sum of interior angles of a triangle formula