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A quadrilateral such as BCEF is called an adventitious quadrangle when the angles between its diagonals and sides are all rational angles, angles that give rational numbers when measured in degrees or other units for which the whole circle is a rational number. Numerous adventitious quadrangles beyond the one appearing in Langley's puzzle have ...
In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. [57] The size of an angle is formalized as an angular measure. In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right. [43]
Thus a single line can be drawn connecting all nine dots—which would appear as three lines in parallel on the paper, when flattened out. [18] It is also possible to fold the paper flat, or to cut the paper into pieces and rearrange it, in such a way that the nine dots lie on a single line in the plane (see fold-and-cut theorem). [17]
Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from the x-axis to this segment. It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line.
Second, if a transversal intersects two lines so that interior angles on the same side of the transversal are supplementary, then the lines are parallel. These follow from the previous proposition by applying the fact that opposite angles of intersecting lines are equal (Prop. 15) and that adjacent angles on a line are supplementary (Prop. 13).
An angle equal to 1 / 4 turn (90° or π / 2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular. [12] An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle [11] ("obtuse" meaning "blunt").
Five room puzzle – Cross each wall of a diagram exactly once with a continuous line. [2] MU puzzle – Transform the string MI to MU according to a set of rules. [3] Mutilated chessboard problem – Place 31 dominoes of size 2×1 on a chessboard with two opposite corners removed. [4] Coloring the edges of the Petersen graph with three colors. [5]
a) different tangent lines (transversal intersection, after transversality), or b) the tangent line in common and they are crossing each other (touching intersection, after tangency). If both the curves have a point S and the tangent line there in common but do not cross each other, they are just touching at point S.