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In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180 ...
If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side. Euclid's parallel postulate states: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles , then the two lines, if extended indefinitely ...
With parallel lines, they are congruent. Alternate angles are the four pairs of angles that: have distinct vertex points, lie on opposite sides of the transversal and; both angles are interior or both angles are exterior. If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent.
In several high school treatments of geometry, the term "exterior angle theorem" has been applied to a different result, [1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This result, which depends upon Euclid's parallel ...
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 ]
The sum of the angles is the same for every triangle. There exists a pair of similar, but not congruent, triangles. Every triangle can be circumscribed. 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.
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 ...
(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]