Section 9-5: Inscribed Angles & Corollaries

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Presentation transcript:

Section 9-5: Inscribed Angles & Corollaries

ÐABC is an inscribed angle of Circle O. Definition: an Inscribed Angle is an angle with its vertex on the circle. A O C B ÐABC is an inscribed angle of Circle O.

This inscribed angle intercepts an arc of Circle O. The intercepted 110° The measure of the intercepted arc of an inscribed angle is equal to twice the measure of the inscribed angle. O C 55° B This inscribed angle intercepts an arc of Circle O. The intercepted arc is AC.

ÐABC and ÐADC both intercept AC. Corollary 1: Inscribed angles that intercept the same arc are congruent. A D C B 100° ÐABC and ÐADC both intercept AC. 50° 50° mÐABC = 50 mÐADC = 50

Corollary 2: An angle inscribed inside of a semicircle is a right angle. 110° 70° mÐBAC = 90 35° 55° Here’s why…

Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. A D C B mÐBAC = 76 76° 92° mÐACD = 92 88° 104°

Section 9-6: Other Angles

Theorem 9-10: The measure of an angle formed by two secants, two tangents, or a secant & a tangent is equal to half the difference of the intercepted arcs. mÐABC = ½(mAC – mDE). A D C B E

Case 1: Two Secants A D C B E 84° 20°

Case 2: Two Tangents mÐADB = ½(226 – 134) A D B mÐADB = 46 226° 134°

Case 3: A secant & a tangent A D C B 30° 128°

mAG = 100 mCE = 30 mEF = 25 A D C B E F O G 3 7 8 2 1 4 5 6 Given: AB is tangent to Circle O; AF is a diameter. Find all numbered angles. A D C B E F O G 3 7 8 2 1 4 5 6 mAG = 100 mCE = 30 mEF = 25

m1 = ___ m2 = ___ m3 = ___ m4 = ___ m5 = ___ m6 = ___ m7 = ___ m8 = ___