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Cairo pentagonal tiling by equilateral pentagons with two non-adjacent right angles. There are two infinite families of equilateral convex pentagons that tile the plane, one having two adjacent supplementary angles and the other having two non-adjacent supplementary angles. Some of those pentagons can tile in more than one way, and there is one ...
It is possible to divide an equilateral triangle into three congruent non-convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three-pentagon unit. [21] A similar method can be used to subdivide squares into four congruent non-convex pentagons, or regular hexagons into six congruent non-convex ...
Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V. Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon. Draw a circle of radius OA and ...
Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26).
In the concave case, the line through one of the diagonals bisects the other.) One diagonal is a line of symmetry. It divides the quadrilateral into two congruent triangles that are mirror images of each other. [7] One diagonal bisects both of the angles at its two ends. [7]
A compound of two "line segment" digons, as the two possible alternations of a square (note the vertex arrangement). The apeirogonal hosohedron , containing infinitely narrow digons. Any straight-sided digon is regular even though it is degenerate, because its two edges are the same length and its two angles are equal (both being zero degrees).
If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be constructed. If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a vertex. Because p and q are coprime, there exists integers a and b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq.
Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle ...