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Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C C A B B B All three of these inscribed angles intercept arc AB. C Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C C A B B B C All three of these inscribed angles intercept arc AB. Theorem : An inscribed angle is equal to half of its intercepted arc. 1 C AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 1 200° 32° C A 2 B 40° B B 3 C Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3. C 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 1 200° 32° C A 2 B 40° B B 3 C Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3. 200 2 1 100 1 C 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 1 200° 32° C A 2 B 40° B B 3 C Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE : Find the measure of angles 1 , 2 and 3. 200 1 2 1 100 40 2 2 2 20 C 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 1 200° 32° C A 2 B 40° B B 3 C Theorem : An inscribed angle is equal to half of its intercepted arc. C EXAMPLE : Find the measure of angles 1 , 2 and 3. 200 1 2 1 100 40 2 2 2 20 32 3 2 3 16 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 86° ? ? C A 25° 18° ? B B C Theorem : An inscribed angle is equal to half of its intercepted arc. EXAMPLE #2 : Find the measure of arc AB in each example. B C 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 86° ? ? C A 25° 18° ? B B C Theorem : An inscribed angle is equal to half of its intercepted arc. C EXAMPLE #2 : Find the measure of arc AB in each example. 1 AB 2 1 862 2 AB 2 172 AB B 86 Take notice that the arc is two time bigger than the angle. 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 86° ? ? C A 25° 18° ? B B C Theorem : An inscribed angle is equal to half of its intercepted arc. C EXAMPLE #2 : Find the measure of arc AB in each example. 1 AB 2 1 862 2 AB 2 172 AB 86 B 2 C AB Take notice that the arc is two times bigger than the angle. 1 AB 2 Geometry – Inscribed and Other Angles Inscribed Angle – an angle whose vertex lies on the circle and its sides are chords of the circle. A A C 86° ? 36° C A 25° 18° 50° B B B C Theorem : An inscribed angle is equal to half of its intercepted arc. C 1 AB 2 EXAMPLE #2 : Find the measure of arc AB in each example. 1 AB 2 1 862 2 AB 2 172 AB 86 2 C AB 2 25 50 2 18 36 Take notice that the arc is two times bigger than the angle. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. A 1 B 1 1 AB 2 Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. A 1 1 AB 2 1 B EXAMPLE : If arc AB = 65°, find the measure of angle 1. Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. A 1 1 AB 2 1 B EXAMPLE : If arc AB = 65°, find the measure of angle 1. 1 1 65 2 1 32.5 Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. X A 1 1 AB 2 1 B EXAMPLE #2 : If arc AXB = 300°, find the measure of angle 1. AB 360 AXB AB 360 300 AB 60 Geometry – Inscribed and Other Angles Theorem : An angle formed by a tangent line and a chord is equal to half of its intercepted arc. X A 1 1 AB 2 1 B EXAMPLE #2 : If arc AXB = 300°, find the measure of angle 1. AB 360 AXB AB 360 300 AB 60 1 1 60 2 1 30 Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C A X B D 1 CXA AC BD 2 Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C 40° A X B 42° 1 CXA AC BD 2 D EXAMPLE : Arc AC = 40° and arc BD = 42°. Find the measure of angle CXA. 1 40 42 2 1 CXA 82 2 CXA 41 CXA Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C ? A X B 50° 1 CXA AC BD 2 D EXAMPLE # 2 : Angle CXA = 40° and arc BD = 50°. Find the measure of arc CA. Geometry – Inscribed and Other Angles Theorem : An angle formed by two chords is equal to half of the sum of the intercepted arcs C y A X B 50° 1 CXA AC BD 2 D EXAMPLE # 2 : Angle CXA = 40° and arc BD = 50°. Find the measure of arc CA. 1 40 50 y 2 1 402 50 y 2 2 80 50 y 30 y Geometry – Inscribed and Other Angles Theorem : An angle formed by two secants is equal to half of the difference of the intercepted arcs. ( a secant is a line that cuts through a circle ) D C X A B 1 X BD AC 2 Geometry – Inscribed and Other Angles Theorem : An angle formed by two secants is equal to half of the difference of the intercepted arcs. ( a secant is a line that cuts through a circle ) D C 75° 23° A X 1 X BD AC 2 B EXAMPLE : Arc BD = 75° and arc CA = 23°. Find the measure of angle “x” . Geometry – Inscribed and Other Angles Theorem : An angle formed by two secants is equal to half of the difference of the intercepted arcs. ( a secant is a line that cuts through a circle ) D C 75° 23° A X 1 X BD AC 2 B EXAMPLE : Arc BD = 75° and arc CA = 23°. Find the measure of angle “x” . 1 X 75 23 2 1 X 52 2 X 26