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 Angle Y Z O 110 Intercepted Arc

EXAMPLE Segment AD is a diameter. Find the values of x and y and z in the figure. x = 25° y = 100° z = 55° A B O C D 55 x y 25 z

SUM OF CENTRAL ANGLES The sum of the measures fo the central angles of a circle with no interior points in common is 360º. 360º

Find the measure of each arc. 2x-14 4x 2x 3x E B 3x+10 4x + 3x + 3x + 10+ 2x + 2x – 14 = 360 … x = 26 104, 78, 88, 52, 66 degrees A

Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords. 3 1 2 4 Is NOT! Is SO! Is NOT! Is SO!

INSCRIBED ANGLE THEOREM Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Two inscribed angles that intercept the same arc are congruent.

INSCRIBED ANGLE THEOREM Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.

INSCRIBED ANGLE THEOREM Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Inscribed Angle Y 110 55 Z Intercepted Arc

Find the value of x and y in the figure. Thrm 9-7. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Find the value of x and y in the figure. X = 20° Y = 60° P 40 Q 50 y S x R T

Find the value of x and y in the figure. Corollary 1. If two inscribed angles intercept the same arc, then the angles are congruent.. Find the value of x and y in the figure. X = 50° Y = 50° P Q y 50 S R x T

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 mPRQ = ½ mPR

What is mPRQ ? P Q 60 S R

Chord Central Angles Conjecture P S R If two chords are congruent, then their central angles and intercepted arcs are congruent.

An angle inscribed in a semicircle is a right angle. P 180 90 S 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 ½ 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 ½ the sum of the measures of the intercepted arcs. 1 A B C D

Interior Angle Theorem 91 A C x° y° B D 85

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.)

A circumscribed angle is formed by two lines tangent to the circle. P

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 j 1 k 1 k j 1

Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs. 1 j k 3

Find <C = ½(265-95) <C = ½(170) m<C = 85°

PUTTING IT TOGETHER! AF is a diameter. mAG=100 mCE=30 mEF=25 Find the measure of all numbered angles. Q G F D E C 1 2 3 4 5 6 A

Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. P Q mPSR + mPQR = 180  S R