Friday-Chapter 6 Quiz 2 on 6.1-6.3 Lesson 6.3 Inscribed Angles and their Intercepted Arcs Objectives: Using Inscribed Angles Using Properties of Inscribed Angles. Homework: Lesson 6.3/ 1-12 Friday-Chapter 6 Quiz 2 on 6.1-6.3

Inscribed Angles & Intercepted Arcs Using Inscribed Angles 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

Measure of an Inscribed Angle Using Inscribed Angles Measure of an Inscribed Angle  

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

Q R Find the measure of each arc or angle. = ½ 120 = 60˚ = 180˚ 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˚

Inscribed Angles Intercepting Arcs Conjecture 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˚

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

Cyclic Quadrilateral Quadrilateral ABFE is inscribed in Circle O. 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.

Cyclic Quadrilateral Conjecture 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°

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

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

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

Circumscribed Polygon 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.

Angles inscribed in a Semi-circle Conjecture 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 the diameter is the hypotenuse           A has its vertex on the circle, and it intercepts half of the circle so that mA = 90.

Example 8: Angles inscribed in a semi-circle are right angles Find x.    

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

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

Using Properties of Inscribed Angles Example 10: Find x. x 122˚ 189˚ 360 – 189 – 122 = 49˚ x = 49/2 = 24.5˚ x

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

Example 11:            

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

Homework: Lesson 6.3/ 1-12