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INSCRIBED ANGLE

This content explores inscribed angles, defined as angles with their vertex on the circle and sides as chords. Key concepts include the relationship between the inscribed angle and the intercepted arc, where the inscribed angle measures half the intercepted arc. Several examples illustrate how to find angle measures and the values of variables in equations related to inscribed angles. The document also discusses how a right triangle inscribed in a circle has its hypotenuse as the circle's diameter and outlines properties of cyclic quadrilaterals.

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INSCRIBED ANGLE

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

  2. 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 intercepted arc!! 120 x y

  3. J K Q S M Examples 1. If m JK = 80, find m JMK. 40  2. If mMKS = 56, find m MS. 112 

  4. 3. Find the measure of arc AL. (think about it!) F L K 48º A

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

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

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

  8. Example 5 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

  9. Example 6 In K, m  1 = 6x – 5 and m2  = 3x – 4. Find the value of x. 6x – 5 + 3x – 4 = 90 H 2 K x = 11 N 1 G

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

  11. Example 7 Find y and z. 110 + y+6 =180 z 110 y+6 y = 64 85 z + 85 = 180 z = 95

  12. Homework: Pg 207-208 1-20 all!

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