Inscribed Angles in Geometry
Explore the concept of inscribed angles in geometry, learn how to use them to solve problems, and understand the theorems related to inscribed angles such as the Measure of an Inscribed Angle Theorem and Congruent Inscribed Angles Theorem. Discover how inscribed angles are related to circles, arcs, and polygons, and delve into practical applications of inscribed angles in real-life scenarios.
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Presentation Transcript
Inscribed Angles Section 10.5
Essential Questions 1. How to use inscribed angles to solve problems in geometry 2. How to use inscribed angles to solve real-life problems
Definitions Inscribed angle an angle whose vertex is on a circle and whose sides contain chords of the circle Intercepted arc the arc that lies in the interior of an inscribed angle and has endpoints on the angle intercepted arc inscribed angle
Measure of an Inscribed Angle Theorem (10.9) If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. A C m ADB = 1 2mAB D B
Example 1 Find the measure of the blue arc or angle. E a. S b. R 80 F G Q T m EFG = 1 2(80 ) = 40 mQTS = 2(90 ) = 180
Congruent Inscribed Angles Theorem (10.10) If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A B C D C D
Example 2 It is given that m E = 75 . What is m F? Since E and F both intercept the same arc, we know that the angles must be congruent. D E m F = 75 F H
Definitions Inscribed polygon a polygon whose vertices all lie on a circle. Circumscribed circle A circle with an inscribed polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.
Inscribed Right Triangle Theorem(10.11) 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. A B is a right angle if and only if AC is a diameter of the circle. B C
Inscribed Quadrilateral Theorem(10.12) A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. E F C D G D, E, F, and G lie on some circle, C if and only if m D + m F = 180 and m E + m G = 180 .
Example 3 Find the value of each variable. D b. a. B z G E y 120 Q A 80 2x F C m G + m E = 180 m D + m F = 180 2x = 90 x = 45 y + 120 = 180 y = 60 z + 80 = 180 z = 100
Assignment: P. 508/ 1- 14 HW: p. 508/ 15- 33 all
