The Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent. Knowing the triangle's parts, here is the challenge: how do we prove that the base angles are congruent? That is the heart of the Isosceles Triangle Theorem, which is built as a conditional (if, then) statement: The two angles formed between base and legs, ∠DUK and ∠DKU, or ∠D and ∠K for short, are called base angles. The third side is called the base (even when the triangle is not sitting on that side). ∠DU ≅ ∠DK, so we refer to those twins as legs. Like any triangle, △DUK has three sides: DU, UK, and DK Like any triangle, △DUK has three interior angles: ∠D, ∠U, and ∠K What else have you got? Properties of an isosceles triangle If these two sides, called legs, are equal, then this is an isosceles triangle. Hash marks show sides ∠DU ≅ ∠DK, which is your tip-off that you have an isosceles triangle. You can draw one yourself, using △DUK as a model. By working through everything above, we have proven true the converse (opposite) of the Isosceles Triangle Theorem.Here we have on display the majestic isosceles triangle, △DUK. When the triangles are proven to be congruent, the parts of the triangles are also congruent making EF congruent with EH. That gives us two angles and a side, which is the AAS theorem. We now have what’s known as the Angle Angle Side Theorem, or AAS Theorem, which states that two triangles are equal if two sides and the angle between them are equal. Because we have an angle bisector with the line segment EG, FEG is congruent with HEG. Label this point on the base as G.īy doing this, we have made two right triangles, EFG and EGH. To do that, draw a line from FEH (E is the apex angle) to the base FH. We need to prove that EF is congruent with EH. The EFH angle is congruent with the EHF angle. It states, “if two angles of a triangle are congruent, the sides opposite to these angles are congruent.” Let’s work through it.įirst, we’ll need another isosceles triangle, EFH. They are visible on flags, heraldry, and in religious symbols.Īs with most mathematical theorems, there is a reverse of the Isosceles Triangle Theorem (usually referred to as the converse). You can also see isosceles triangles in the work of artists and designers going back to the Neolithic era. In the Middle Ages, architects used what is called the Egyptian isosceles triangle, or an acute isosceles triangle. Ancient Greeks used obtuse isosceles triangles as the shapes of gables and pediments. Ancient Egyptians used them to create pyramids. Īs far as isosceles triangles, you see them in architecture, from ancient to modern. You can also see triangular building designs in Norway, the Flatiron Building in New York, public buildings and colleges, and modern home designs. The triangular shape could withstand earthquake forces, unlike a rectangular or square design. In 1989, Japanese architects decided that a triangular building design would be necessary if they were to construct a 500-story building in Tokyo. With modern technology, triangles are easier to incorporate into building designs and are becoming more prevalent as a result. While rectangles are more prevalent in architecture because they are easy to stack and organize, triangles provide more strength.
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