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If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent. If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of ...
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 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]
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 ...
In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180 ...
the pattern is the same. picture retouched: removed black background, made triangle shading uniform. 08:25, 6 June 2011: 1,275 × 1,650 (59 KB) Lsalgo {{Information |Description ={{en|1=Using a variant of Lubachevsky-Stillinger algorithm, 1000 congruent equilateral triangles are randomly packed in a rectangle with periodic (wrap-around) boundary.
Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined. [13]: pp. 201–203
All ideal triangles are congruent to each other. The interior angles of an ideal triangle are all zero. An ideal triangle has infinite perimeter. An ideal triangle is the largest possible triangle in hyperbolic geometry. In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties: