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In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), [1] [2] is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could ...
Angle of parallelism in hyperbolic geometry. In hyperbolic geometry, angle of parallelism () is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.
If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. There exists a quadrilateral in which all angles are right angles, that is, a rectangle. There exists a pair of straight lines that are at constant distance from each other. Two lines that are parallel to the same line are also parallel to each other.
An acute trapezoid has two adjacent acute angles on its longer base edge. An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base. An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry. This is ...
Saccheri quadrilaterals. A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base.It is named after Giovanni Gerolamo Saccheri, who used it extensively in his 1733 book Euclides ab omni naevo vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum.
(In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares.) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle. [2]: p. 115 All triangles in which the Euler line is parallel to one side are acute. [3]
In the picture below, angles ∠ABC and ∠DCB are obtuse angles of the same measure, while angles ∠BAD and ∠CDA are acute angles, also of the same measure. Since the lines AD and BC are parallel, angles adjacent to opposite bases are supplementary, that is, angles ∠ABC + ∠BAD = 180°.
The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a 45-degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it.