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Understanding Inscribed Angles and Arc Relationships in Circles

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Dive into the concepts of central angles, inscribed angles, and their relationships within circles. Learn that the measure of an inscribed angle is half that of its intercepted arc. Explore examples that illustrate how to calculate unknown angles using given measurements and the properties of inscribed and central angles. Understand how inscribed polygons relate to circumscribed circles, including properties of supplementary angles in quadrilaterals and triangles. This guide offers clarity on geometric relationships essential for mastering circle theorems.

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Understanding Inscribed Angles and Arc Relationships in Circles

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Presentation Transcript

  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

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