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Transcript
11.3 Inscribed angles By: Mauro and Pato Objectives: • Define Inscribe angle, intercepted arc, inscribed and circumscribed. • Be able to know the theorems “Measure of an Inscribed Angle”, Theorem 10.9 and theorems about inscribed polygons (theorem 10.10 and 10.11). • Be able to apply this knowledge on any kind of problems related to this topic. • Laugh the 45 minutes of class and have a good time. Geometry Background: • Circles: a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center). • Polygons: a plane figure with at least three straight sides and angles, and typically five or more. • Supplementary Angles: either of two angles whose sum is 180º. You will learn: • What is an inscribed angle. • What is an intercepted arc. • Four different theorems and how to apply them. Inscribed Angle Measure of an Inscribed Angle • If an angle is inscribed in a circle, then its measure is half its measure is half the measure of its intercepted arc A • m<ADB = ½ mAB D B example Theorem 10.9 • If two inscribed angles of a circle intercept the same arc, then the angles are congruent A B C D example Remember? • Circumscribed: (draw a figure around another) touching the figure by touching its sides but not cutting it. • Inscribed: draw a figure within another so that their boundaries touch but do not intersect. Theorem 10.10 • If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. C A B example Theorem 10.11 • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. E F C D G example