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Understanding Inscribed Angles and Arc Measures in Geometry

This lesson focuses on the properties of inscribed angles and their relationship to intercepted arcs in circles. We explore how to locate a gazebo among three houses using geometric centers, identify inscribed angles, and calculate their measures. Students will learn that the measure of an inscribed angle is half the measure of its intercepted arc. The homework will consist of practice problems to reinforce these concepts, including the relationship between angles and arcs, and properties of inscribed and circumscribed polygons.

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Understanding Inscribed Angles and Arc Measures in Geometry

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  1. Warm-up Find the measure of each arc.

  2. Homework Review

  3. A gazebo is to be constructed between three houses so that it will approximately be the same distance from each house.  Which would be helpful in deciding where to locate the gazebo? EOCT Review a) Centroid b) Circumenter c) Incenter d) Orthocenter b

  4. CCGPS GeometryDay 21 (9-5-13) UNIT QUESTION: What special properties are found with the parts of a circle? Standard: MMC9-12.G.C.1-5,G.GMD.1-3 Today’s Question: How do we use angle measures to find measures of arcs? Standard: MMC9-12.G.C.2

  5. Inscribed Angles

  6. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle INTERCEPTEDARC INSCRIBEDANGLE

  7. YES; CL C T O L Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 1.

  8. NO; QVR Determine whether each angle is an inscribed angle. Name the intercepted arc for the angle. 2. Q V K R S

  9. To find the measure of an inscribed angle… 160 80

  10. What do we call this type of angle? What is the value of x? What do we call this type of angle? How do we solve for y? The measure of the inscribed angle is HALF the measure of the inscribed arc!! 120 x y

  11. J K Q S M Examples 3. If m JK = 80, find mJMK. 40  4. If mMKS = 56, find m MS. 112 

  12. If two inscribed angles intercept the same arc, then they are congruent. 72

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

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

  15. 2 Column Proof

  16. A circle can be circumscribed around a quadrilateral if and only if its opposite angles are supplementary. B A D C

  17. Example 8 Find y and z. z 110 110 + y =180 y y = 70 85 z + 85 = 180 z = 95

  18. If a right triangle is inscribed in a circle then the hypotenuse is the diameter of the circle. 180 diameter

  19. Example 6 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

  20. Example 7 In K, m1 = 6x – 5 and m2 = 3x – 4. Find the value of x. 6x – 5 + 3x – 4 = 90 H 2 K x = 11 N 1 G

  21. Practice Worksheet

  22. Homework: • Worksheet

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