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English: This diagram illustrates the geometric principle of angle-angle-side triangle congruence: Given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if angle CAB is congruent with C'A'B' and angle BCA is congruent with B'C'A' and BC is congruent with B'C'.
Construct the orthocenter of triangle and three midpoints (say A', B' C' ) between vertices and orthocenter. Construct a circumcircle of A'B'C' . This is the nine-point circle , it intersects each side of the original triangle at two points: the base of altitude and midpoint.
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The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles). [9] The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles. [10]
Congruence [ edit ] If A , B are two points on a line a , and if A ′ is a point upon the same or another line a ′, then, upon a given side of A ′ on the straight line a ′, we can always find a point B ′ so that the segment AB is congruent to the segment A ′ B ′.
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]
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]