Inscribed Angles and their Intercepted Arcs Objectives Using

Inscribed Angles and their Intercepted Arcs Objectives: Using Inscribed Angles Using Properties of Inscribed Angles. 1

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

Using Inscribed Angles Measure of an Inscribed Angle 3

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

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˚ 5

Using 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 6

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

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

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. 9

Using Properties of Inscribed Angles Cyclic Quadrilateral Conjecture If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. m∠A + m∠C = 180° m∠B + m∠D = 180° 10

Using Properties of Inscribed Angles Example 5: Find the measure of Opposite angles of an inscribed quadrilateral are supplementary Intercepted arc of an inscribed angles = 2* angle measure 11

Example 6: Find m∠A and m∠B Opposite angles of an inscribed quadrilateral are supplementary m∠A + 60° = 180° m∠A = 120° m∠B + 140° = 180° m∠B = 40° 12

Example 7: Using Properties of Inscribed Angles Find x and y Opposite angles of an inscribed quadrilateral are supplementary 13

Using Properties of Inscribed Angles Circumscribed Polygon A polygon is circumscribed about a circle if and only if each side of the polygon is tangent to the circle. 14

Using Properties of Inscribed Angles inscribed in a Semi-circle Conjecture A triangle inscribed in a circle is a right triangle if and only if the diameter is the hypotenuse A has its vertex on the circle, and it intercepts half of the circle so that m A = 90. 15

Example 8: Angles inscribed in a semicircle are right angles Find x. 16

Using Inscribed Angles Example 9: Find 146° m FDE 17

Using Properties of Inscribed Angles Parallel (Secant) Lines Intercepted Arcs Conjecture Parallel (secant) lines intercept congruent arcs. A X Y B 18

Example 10: Using Properties of Inscribed Angles Find x. x 18 9˚ 360 – 189 – 122 = 49˚ 12 2˚ x = 49/2 = 24. 5˚ x 19

Tangent/Chord Conjecture The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. B D C C

Example 11: 21

Example 12: Using Tangent/Chord Conjecture Find x and y. Triangle sum 90 o Q J 35 o yo 55 xoo L K

Homework: Lesson 6. 3/ 1 -12 23

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