Since a triangle is obtuse or right if and only if one of its angles is obtuse or right, respectively, an isosceles triangle is obtuse, right or acute if and only if its apex angle is respectively obtuse, right or acute. In Euclidean geometry, the base angles can not be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Whether an isosceles triangle is acute, right or obtuse depends only on the angle at its apex.
In the equilateral triangle case, since all sides are equal, any side can be called the base. The vertex opposite the base is called the apex. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles.
In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The same word is used, for instance, for isosceles trapezoids, trapezoids with two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle. "Isosceles" is made from the Greek roots "isos" (equal) and "skelos" (leg). A triangle that is not isosceles (having three unequal sides) is called scalene. The difference between these two definitions is that the modern version makes equilateral triangles (with three equal sides) a special case of isosceles triangles. Terminology, classification, and examples Įuclid defined an isosceles triangle as a triangle with exactly two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base.Įvery isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two equal sides are called the legs and the third side is called the base of the triangle. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.Įxamples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. In geometry, an isosceles triangle ( / aɪ ˈ s ɒ s ə l iː z/) is a triangle that has two sides of equal length. But if a pentagon is an obtuse triangle glued to a rectangle, then the pentagon has two other obtuse angles in addition to the one we used, and using either one of these other obtuse angles, we get a quadrilateral that can't be a rectangle.Isosceles triangle with vertical axis of symmetry Any convex $n$-gon with $n\ge4$, except the rectangle, has at least one obtuse angle cutting off the triangle containing this obtuse angle and the two adjacent vertices yields a convex $n-1$-gon, so induction yields the claimed result, provided that when going from a pentagon to a quadrilateral we can avoid forming a rectangle. Every convex $n$-gon, $n\ge5$, has one or more obtuse angles, which we can use to cut off triangles, to reduce the $3n-6$ further.ĮDIT ––– Taking this observation to its logical conclusion, we can see that any convex $n$-gon, other than a rectangle, can be partitioned into $n$ obtuse triangles (a rectangle can be partitioned into six obtuse triangles).
E.g., if a convex quadrangle is not a rectangle, then it has at least one obtuse angle, so we can cut off an obtuse triangle incorporating that angle, and just need three more triangles to finish the job, four triangles in all. A convex $n$-gon can be cut into $n-2$ triangles by just choosing a vertex and drawing all the diagonals from that vertex, so $3n$ obtuse triangles can be reduced to $3n-6$.
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