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In a hyperbolic triangle the sum of the angles A, B, C (respectively opposite to the side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle.
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
2: 32 "The Worms Return" Area 3: 33 "A Very Fine Pet is a Polyhedron" 3-D shapes 4: 34 "Better get back in Shape and be a Square" Classifying quadrilaterals 5: 35 "Angleman!" Measuring angles 6: 36 "The X and Y Files" plotting coordinates 7: 37 "Makeover" Recognising translations 8: 38 "Turning Points" Recognising rotations 9: 39 "On Reflection ...
Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A degenerate triangle , whose vertices are collinear , has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of ...
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
[2] The general theory of angle can be unified with invariant measure of area. The hyperbolic angle is defined in terms of area, very nearly the area associated with natural logarithm. The circular angle also has area interpretation when referred to a circle with radius equal to the square root of two.
The defect of any of the vertices of a regular dodecahedron (in which three regular pentagons meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.
The direct theorem was Proposition 22 in Book 3 of Euclid's Elements. [3] 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 ...