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 ExamplesĮuclid 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.ġ. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. 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. 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. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. 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. If you know that two base angles are congruent, then it has to be an isosceles triangle with two sides that are congruent.In geometry, an isosceles triangle (/aɪˈsɒsəliːz/) is a triangle that has at least two sides of equal length. Can we assume that this is an isosceles triangle where two sides are congruent? The answer to that is yes. What about the converse? Remember the converse is when you switch the if and the then part of a conditional statement, so over here I've drawn a triangle where you have two base angles that are congruent. If you're interested, you can also call this side here that's opposite the vertex the base. Our base angles are the two angles that are not part of the vertex but wait a minute, what's the vertex? Well the vertex is this angle right here and notice that the vertex contains the two sides of the triangle that are congruent, so what I said is if you know that two sides of a triangle are congruent then the two base angles must be congruent. If we know that we can assume that the base angles are congruent but wait a minute I just used a couple new words.įirst one was base angles. The first is isosceles triangles mean we have two sides that are congruent. When we're talking about isosceles triangles there are two key things.
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