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

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

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

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˚

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

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

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

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

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.

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

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

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

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.

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

15 Using Properties of Inscribed Angles Find x and y

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

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

18 Homework: Lesson 6.3/ 1-14