Lesson 6.3 Inscribed Angles and their Intercepted Arcs

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

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

Using Inscribed Angles Measure of an Inscribed Angle 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

Using Inscribed Angles Example 1: Find the mand mPAQ . 63° =2 * m PBQ = 2 * 63 = 126˚ mPAQ = m PBQ mPAQ = 63˚

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

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

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

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

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.

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

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

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

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

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

Using Properties of Inscribed Angles Find x and y

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

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

Homework: Lesson 6.3/ 1-14

Lesson 6.3 Inscribed Angles and their Intercepted Arcs