Geometry For Dummies
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An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article discusses the three types of angles that have their vertex outside a circle: secant-secant angles, secant-tangent angles, and tangent-tangent angles. A tangent is a line that touches a circle at a single point; a secant is a line that intersects a circle at two points.

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So here are the three types of angles that have their vertex outside a circle:

  • Secant-secant angle: A secant-secant angle, like angle BDF in the above figure on the top left, is an angle whose vertex lies outside a circle and whose sides are two secants of the circle.

  • Secant-tangent angle: A secant-tangent angle, like angle GJK in the above figure on the top right, is an angle whose vertex lies outside a circle and whose sides are a secant and a tangent of the circle.

  • Tangent-tangent angle: A tangent-tangent angle, like angle LMN in the above figure on the bottom, is an angle whose vertex lies outside a circle and whose sides are two tangents of the circle.

Measure of an angle outside a circle: The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle is one-half the difference of the measures of the intercepted arcs. For example, in the above figure,

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Note that you subtract the smaller arc from the larger (if you get a negative answer, you know you subtracted in the wrong order).

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Now, using the above figure, solve this problem with the angle-outside-a-circle formula:

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Plug these expressions into the formula, and you’re home free:

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About This Article

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About the book author:

Mark Ryan has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses and standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics. Ryan is the author of Calculus For Dummies, Calculus Workbook For Dummies, and Geometry Workbook For Dummies.

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