8-5 Angles in Circles Central Angles A central angle is an angle whose vertex is the CENTER of the circle Central Angle (of a circle) Central Angle (of a circle) NOT A Central
Angle (of a circle) CENTRAL ANGLES AND ARCS The measure of a central angle is equal to the measure of the intercepted arc. CENTRAL ANGLES AND ARCS The measure of a central angle is equal to the measure of the intercepted arc. Central Y Angle 110 0
11 O Z Intercepted Arc EXAMPLE Segment AD is a diameter. Find the values of x and y and z in the figure. 25 B C
A x y O z 55 D x = 25 y = 100 z = 55 SUM OF CENTRAL ANGLES The sum of the measures fo the central angles of a circle with
no interior points in common is 360. 360 Find the measure of each arc. D 2x -14 C 4x 3x
0 +1 3x E 2x 4x + 3x + 3x + 10+ 2x + 2x 14 = 360 x = 26 A 104, 78, 88, 52, 66 degrees B
Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords. 1 Is NOT! 2 Is SO! 3 Is
NOT! 4 Is SO! Thrm 9-7. The measure of an inscribed angle is INSCRIBED ANGLE THEOREM equal to the measure of the intercepted arc. The measure of an inscribed angle is equal to the measure of the intercepted arc.
Thrm 9-7. The measure of an inscribed angle is INSCRIBED ANGLE THEOREM equal to the measure of the intercepted arc. The measure of an inscribed angle is equal to the measure of the intercepted arc. 1 2
Thrm 9-7. The measure of an inscribed angle is INSCRIBED ANGLE THEOREM equal to the measure of the intercepted arc. The measure of an inscribed angle is equal to the measure of the intercepted arc. Inscribed Angle 0 11 55
Y Z Intercepted Arc Thrm 9-7. Thethe measure of an Find value ofinscribed x and angle y is equal to the measure of the intercepted arc.
in the figure. X = 20 P 40 Q S 50 y x T
R Y = 60 Corollary 1. Ifthe two inscribed intercept Find value ofangles x and y the same arc, then the angles are congruent.. in the figure.
P y X = 50 Q Y = 50 S 50 x T R
An angle formed by a chord and a tangent can be considered an inscribed angle. An angle formed by a chord and a tangent can be considered an inscribed angle. P Q S R mPRQ = mPR What is mPRQ ? P
Q S 60 R An angle inscribed in a semicircle is a right angle. P 180 R
An angle inscribed in a semicircle is a right angle. P S 180 90 R Interior Angles Angles that are formed by two intersecting chords. (Vertex IN the circle) A
D B C Interior Angle Theorem The measure of the angle formed by the two chords is equal to the sum of the measures of the intercepted arcs. Interior Angle Theorem The measure of the angle formed by the two chords is equal to the sum of the measures of the intercepted arcs. A
D 1 B C 1 m1 (mAC mBD) 2 Interior Angle Theorem A 91 C
y x B D 85 1 x (91 85) 2 x 88 y 180 88 y 92
Exterior Angles An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. (vertex OUT of the circle.) Exterior Angles An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. k j 1 k
j 1 k j 1 Exterior Angle Theorem The measure of the angle formed is equal to the difference of the intercepted arcs. k j
1 k j 1 1 m1 (k j) 2 k j
3 Find mACB
PUTTING IT TOGETHER! D 6 C E A 3 Q 2 1 5
4 G F AF is a diameter. mAG=100 mCE=30 mEF=25
Find the measure of all numbered angles. Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. P Q mPSR + mPQR = 180 S R
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