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SSS (side-side-side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent. ASA (angle-side-angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA postulate is attributed to Thales of Miletus.
All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. This is a total of six equalities, but three are often sufficient to prove congruence ...
The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established in that side BC corresponds to (in this case of congruence, equals in length) JK, CD corresponds to KL, DA corresponds to LI, and AB corresponds to IJ, while angle ∠C corresponds to (equals) angle ∠K, ∠D corresponds to ∠L, ∠A ...
In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. [1]
The Sierpiński triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets: Start with an equilateral triangle. Subdivide it into four smaller congruent equilateral triangles and remove the central triangle. Repeat step 2 with each of the remaining smaller triangles infinitely.
There are several elementary results concerning similar triangles in Euclidean geometry: [9] Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other (transitivity of similarity of triangles). Corresponding altitudes of similar triangles have the same ratio as the corresponding sides.
In other words, all of its faces are the same size and shape (congruent) and all edges are the same length. The regular tetrahedron is the simplest convex deltahedron, a polyhedron in which all of its faces are equilateral triangles; there are seven other convex deltahedra. [3]
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