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Understanding Central and Inscribed Angles in Circles: Key Concepts and Examples

This review covers essential concepts of central and inscribed angles in circles. It explains how the vertex position determines the angle's properties, illustrating that the arc of a circle is twice as large as its inscribed angle. Through multiple examples, we investigate different scenarios involving angle measures, including finding unknown variables. Additionally, we emphasize the properties of inscribed polygons, elaborating on supplementary angles and the significance of diameters in right triangles. This guide is perfect for reinforcing comprehension through practice problems.

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Understanding Central and Inscribed Angles in Circles: Key Concepts and Examples

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  1. Warm up April 22

  2. EOCT Week 14 #2

  3. Review HW

  4. Case I:Central Angle: Vertex is ATthe center A  P B

  5. Case II:Inscribed Angle: Vertex is ON circle  ANGLE ARC ARC  ANGLE

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

  7. Find the value of x and y 120  x  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. a 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 Bonus: What type of triangle is this? Why?

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

  16. Inscribed Quadrilaterals

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

  18. Vocab • inscribed quadrilateral a quadrilateral whose vertices are on a circle • inscribed triangle a triangle whose vertices are tangent to a circle

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

  20. 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

  21. a quadrilateral inscribed in a circle: opposite angles are supplementary. B A D C

  22. Find y and z z 110 + y =180 110 y = 70 y 85 z + 85 = 180 z = 95

  23. Practice WS Get in your groups please. 12 problems

  24. Homework WS Day 21 Inscribed Angles Choose even or odd problems

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