This article is about the basic geometric shape. In other words, there is only one plane that contains that triangle, and every triangle is contained in some area of triangle pdf. Euclidean spaces this is no longer true.

This article is about triangles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. The latter definition would make all equilateral triangles isosceles triangles. Equivalently, it has all angles of different measure. In a triangle, the pattern is usually no more than 3 ticks.

An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal. Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles since no angles are equal. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees. A right degenerate triangle has collinear vertices, two of which are coincident.

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 such a triangle is therefore equilateral. A triangle, showing exterior angle d. This allows determination of the measure of the third angle of any triangle given the measure of two angles. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. If and only if one pair of corresponding sides of two triangles are in the same proportion as are another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar. If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.

This is a total of six equalities, but three are often sufficient to prove congruence. 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. SSS: Each side of a triangle has the same length as a corresponding side of the other triangle.

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. 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.

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. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side.

A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. In this section just a few of the most commonly encountered constructions are explained. The length of the altitude is the distance between the base and the vertex. The orthocenter lies inside the triangle if and only if the triangle is acute. The incircle is the circle which lies inside the triangle and touches all three sides. The centroid cuts every median in the ratio 2:1, i. The radius of the nine-point circle is half that of the circumcircle.

The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler’s line. There are various standard methods for calculating the length of a side or the measure of an angle. Angles A and B may vary.