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1 Lesson 6.3 Inscribed Angles and their Intercepted Arcs Goal 1 Using Inscribed Angles Goal 2 Using Properties of Inscribed Angles.

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2 Using Inscribed Angles An INSCRIBED ANGLE is an angle whose vertex is on the circle and whose sides each contain chords of a circle. Inscribed Angles & Intercepted Arcs

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3 Using Inscribed Angles If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. m = m arc OR 2 m = m arc Measure of an Inscribed Angle

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4 Using Inscribed Angles Example 1: 63 Find the m and m PAQ. m PAQ = m PBQ m PAQ = 63˚ =2 * m PBQ = 2 * 63 = 126˚

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5 Using Inscribed Angles Find the measure of each arc or angle. Example 2: QQ RR = ½ 120 = 60˚ = 180˚ = ½(180 – 120) = ½ 60 = 30˚

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6 Using Inscribed Angles Inscribed Angles Intercepting Arcs Conjecture If two inscribed angles intercept the same arc or arcs of equal measure then the inscribed angles have equal measure. m CAB = m CDB

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7 Using Inscribed Angles Example 3: Find =360 – 140 = 220˚

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m = 82˚ 8 Using Properties of Inscribed Angles Example 4: Find m CAB and m m CAB = ½ m CAB = 30 ˚ m = 2* 41˚

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9 Using Properties of Inscribed Angles Cyclic Quadrilateral A polygon whose vertices lie on the circle, i.e. a quadrilateral inscribed in a circle. Quadrilateral ABFE is inscribed in Circle O.

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10 Using Properties of Inscribed Angles If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Cyclic Quadrilateral Conjecture

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11 Using Properties of Inscribed Angles A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle. Circumscribed Polygon

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12 Using Inscribed Angles Example 5: Find m EFD m EFD = ½ 180 = 90˚

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13 Using Properties of Inscribed Angles A triangle inscribed in a circle is a right triangle if and only if one of its sides is a diameter. Angles inscribed in a Semi-circle Conjecture A has its vertex on the circle, and it intercepts half of the circle so that m A = 90.

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14 Using Properties of Inscribed Angles Find the measure of Example 6: Find x.

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15 Using Properties of Inscribed Angles Find x and y

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16 Using Properties of Inscribed Angles Parallel Lines Intercepted Arcs Conjecture Parallel lines intercept congruent arcs. A B X Y

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17 Using Properties of Inscribed Angles Find x. x 122˚ 189˚ 360 – 189 – 122 = 49˚ x = 49/2 = 24.5˚

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18 Homework: Lesson 6.3/ 1-14

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