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Understanding Inscribed Angles and Central Angles in Geometry

Explore the fascinating relationship between inscribed angles and central angles in circles! This guide covers key concepts including the inscribed angle theorem, where the measure of an inscribed angle is half that of the central angle subtending the same arc. Learn how to calculate unknown angles in triangles inscribed in circles, and the properties of inscribed polygons. With numerous examples and practice problems, you will reinforce your understanding of these important geometric principles. Ideal for students seeking to master the topic in geometry.

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Understanding Inscribed Angles and Central Angles in Geometry

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  1. Warm up 30 80 100 180 100 260

  2. Wheel of Formulas!!

  3. Central Angle Angle = Arc

  4. Inscribed Angle • Angle where the vertex is ON the circle

  5. Inscribed Angle

  6. 160 The arc is twice as big as the angle!! 80

  7. Find the value of x and y. 120  120 x  60 y 

  8. J K Q S M Examples 1. If mJK= 80 and JMK = 2x – 4, find x. x = 22 2. If mMKS= 56, find m MS. 112

  9. Find the measure of DOG and DIG D 72˚ G If two inscribed angles intercept the same arc, then they are congruent. O I

  10. If all the vertices of a polygon touch the edge of the circle, the polygon is INSCRIBED and the circle is CIRCUMSCRIBED.

  11. Quadrilateral inscribed in a circle: opposite angles are SUPPLEMENTARY B A D C

  12. If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle. diameter

  13. Q D 3 J T 4 U Example 3 In J, m3 = 5x and m 4 = 2x + 9. Find the value of x. x = 3

  14. Example 4 In K, GH is a diameter and mGNH = 4x – 14. Find the value of x. 4x – 14 = 90 H K x = 26 N G

  15. Example 5 Find y and z. z 110 110 + y =180 y y = 70 85 z + 85 = 180 z = 95

  16. Practice WS 10 problems

  17. Homework WS 6.4 Practice

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