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. The vertex angle Y of triangle XYZ equals 8.57 degrees.Isosceles triangle with vertical axis of symmetry Since we know that X = Z because it is an isosceles triangle, then we can solve for the measures of all the angles. First we read "The degree measure of a base angle", so let's start with X= We need to make an equation out of this problem, so let's figure out what it's trying to tell us. Notice that it's hard to draw a picture without knowing which angles are largest. Find the degree measure of the vertex angle Y. The degree measure of a base angle of isosceles triangle XYZ exceeds three times the degrees measure of the vertex Y by 60. The measure of vertex angle S in triangle RST is 52 degrees. Find the degree measure of the vertex angle S.īase angle + base angle + vertex angle S = 180 degreesĦ4 degrees + 64 degrees + x = 180 degrees Base angles R and T both measure 64 degrees. In isosceles triangle RST, angle S is the vertex angle. (1) Let x = the measure of each base angle.īase angle + base angle + 120 degrees = 180 degreesĮach base angle of triangle ABC measures 30 degrees. Find the degree measure of each base angle. The vertex angle B of isosceles triangle ABC is 120 degrees. The angle located opposite the base is called the vertex. In an isosceles triangle, we have two sides called the legs and a third side called the base. The easiest way to define an isosceles triangle is that it has two equal sides. Similarly, if two angles of a triangle have equal measure, then the sides opposite those angles are the same length. In an isosceles triangle, the base angles have the same degree measure and are, as a result, equal (congruent). There is a special triangle called an isosceles triangle. There are many types of triangles in the world of geometry.
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