Angles In Circles
Made From Tangents, Secants, Radii, Chords

 An inscribed angle consists of two intersecting chords which share an endpoint on the circle. A central angle consists of two radii with a vertex at the circle's center. When a tangent line (BC) intersects with a chord (BA), it forms an angle (∠d) equal to one half of the intercepted arc. When two tangent lines intersect, they form an angle (∠y) equal to:       ½ (the larger intercepted arc minus the smaller intercepted arc) The two tangent lines (xy & zy) will always be of equal length. When a tangent (Line C) and a secant (Lines B + A) intersect, they form an angle (∠y) equal to:       ½ (the larger intercepted arc minus the smaller intercepted arc) The tangent line squared (C2) equals the external secant line segment (A) times the length of the entire secant (A + B). C2 = A • (A + B) When two secants (Line A + B and Line C + D) intersect, they form an angle (∠y) equal to:       ½ (the larger intercepted arc minus the smaller intercepted arc) The line segments they form can be calculated from this formula: (A + B) • B = (C + D) • D Whenever two chords of a circle intersect, they form two pairs of vertical angles: 1) Angles e & f and 2) Angles g & h. Angles e and f, will each equal half the sum of the intercepted arcs: ∠e = ∠f = ½ (arc AC + arc BD) Using the example in the graphic: ∠e = ∠f = ½ (31° + 69°) = 50° We know there are 360° in a circle so (∠g + ∠h) = 360° minus (∠e + ∠f) (∠g + ∠h) = 360° minus (50° + 50°)   (∠g + ∠h) = 260° ∠g = 130° ∠h =130° Whenever two chords intersect, the products of their segments are equal. Looking at the graphic, we see 2 chords intersecting at point "E". When two chords intersect,       chord segments (AE • BE) = (CE • DE) As we can easily see, 8 • 3 = 6 • 4 An inscribed quadrilateral (or a cyclic quadrilateral) has all four of its vertices resting on a single circle. Opposite angles in an inscribed quadrilateral are supplementary (sum to 180°) ∠A + ∠C = 180° ∠B + ∠D = 180°

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