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  2. Transversal (geometry) - Wikipedia

    en.wikipedia.org/wiki/Transversal_(geometry)

    To prove proposition 29 assuming Playfair's axiom, let a transversal cross two parallel lines and suppose that the alternate interior angles are not equal. Draw a third line through the point where the transversal crosses the first line, but with an angle equal to the angle the transversal makes with the second line.

  3. Internal and external angles - Wikipedia

    en.wikipedia.org/wiki/Internal_and_external_angles

    The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...

  4. Absolute geometry - Wikipedia

    en.wikipedia.org/wiki/Absolute_geometry

    (The alternate interior angle theorem states that if lines a and b are cut by a transversal t such that there is a pair of congruent alternate interior angles, then a and b are parallel.) The foregoing construction, and the alternate interior angle theorem, do not depend on the parallel postulate and are therefore valid in absolute geometry. [7]

  5. Parallel postulate - Wikipedia

    en.wikipedia.org/wiki/Parallel_postulate

    If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.

  6. Cyclic quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Cyclic_quadrilateral

    Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle. In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2 n -gon, then the two sums of alternate interior angles are each equal to ( n -1) π {\displaystyle \pi } . [ 4 ]

  7. Angle - Wikipedia

    en.wikipedia.org/wiki/Angle

    Therefore, both angle A and angle B have measures equal to x and are equal in measure. Angles A and B are adjacent. Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm".

  8. Parallelogram - Wikipedia

    en.wikipedia.org/wiki/Parallelogram

    (alternate interior angles are equal in measure). (since these are angles that a transversal makes with parallel lines AB and DC). Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.

  9. Inscribed angle - Wikipedia

    en.wikipedia.org/wiki/Inscribed_angle

    Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore, triangle VOA is isosceles, so angle ∠BVA (the inscribed angle) and angle ∠VAO are equal; let each of them be denoted as ψ. Angles ∠BOA and ∠AOV are supplementary, summing to a straight angle (180°), so angle ∠AOV measures 180° − θ.