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Consecutive interior angles are the two pairs of angles that: [4] [2] have distinct vertex points, lie on the same side of the transversal and; are both interior. Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°).
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 ...
For every concave kite there exist two circles tangent to two of the sides and the extensions of the other two: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle. [27]
A rhombus. Each angle marked with a black dot is a right angle. The height h is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths p and q are the red dotted line segments.
In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2) π radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2) 1 / 2 turn. The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles ...
One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°.. The area of a regular icositetragon is: (with t = edge length) = = (+ + +).
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 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the progression then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°. The right angle ...