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Understanding Inscribed Angles and their Relationships in Circles**

This lesson explores inscribed angles, their measurements, and their relationships with intercepted arcs. Key objectives include finding the measure of an inscribed angle and the angle formed by a tangent and a chord. The Inscribed Angle Theorem states that the measure of an inscribed angle is half that of its intercepted arc. Additional corollaries reveal that angles inscribed in the same arc are congruent, every angle in a semicircle is a right angle, and opposite angles in an inscribed quadrilateral are supplementary. Engage with examples and assignments to solidify your understanding.

bruce-moreno
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Understanding Inscribed Angles and their Relationships in Circles**

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  1. LESSON 11.3 INSCRIBED ANGLES OBJECTIVE: To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord

  2. CR A intercepts what arc? _____ mO = ____ mCR = ____ mA = ____ 60° 60° 30° INTERCEPTED ARC THEOREM 11-9: The measure of an inscribed angle equals the measure of its HALF INTERCEPTED ARC

  3. QP QP A intercepts what arc? _____ B intercepts what arc? _____ mA = ____ mB = ____ 68° 68° Corollary 1: Angles inscribed in the same arc are CONGRUENT

  4. CDE CDE A intercepts what arc? ______ B intercepts what arc? ______ CDE is a ___________ mA = ____ mB = ____ SEMI-CIRCLE 90° 90° Corollary 2: Every angle inscribed in a semicircle is a RIGHT ANGLE

  5. mA = _____ mB = _____ mC = _____ mD = _____ 108° 90° 72° 90° Corollary 3: The opposite angles of a quadrilateral inscribed in a circle are SUPPLEMENTARY

  6. B A D C mA = mBC = mBCD = 120° 120° 60° THEOREM 11-10: The measure of an angle formed by a tangent and a chord is half the measure of its INTERCEPTED ARC

  7. Ex. #1 Find the value of a and b. 60 = ½ a P a 120°= a 60 T Q 30 b = ½ (a + 30) b S b = ½ (120 + 30) R b = ½ (150) An inscribed  is ½ the m of its int. arc. b = 75°

  8. Ex. #2 Find m1. m1 is 90°. 40° 70° 1 ’s inscribed in a semicircle = 90°.

  9. Ex. #3 Find m2. 70° 2 and 38° intercept same arc. 2 38° m2 = 38°. Angles that intcpt the same arc are .

  10. Ex. #4 Find the value of mLJK and y. mLJK = ½ JL J Q ½ JL= mQ = 35° 35 y mLJK = 35° L K s formed by tan.& chord = ½ the intcpt arc.

  11. mQJL = 180° mQJ = 180 - mJL mQJ = 180 - 70 mQJ = 110 y = ½mQJ Ex. #4 Find y. J Q 35 y L K y = 55° An inscribed  is ½ the m of its int. arc.

  12. ASSIGNMENT: Page 601 #1-3, 5-21 write the theorems for 5-21

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