Angles in Circles Central Angles A central angle
- Slides: 28
Angles in Circles
Central Angles • A central angle is an angle whose vertex is the CENTER of the 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 360º
Find the measure of each arc. D 2 x -14 C 4 x 3 x 0 +1 3 x E 2 x 4 x + 3 x + 10+ 2 x – 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 ½ the measure of the intercepted arc. The measure of an inscribed angle is equal to ½ the measure of the intercepted arc.
Thrm 9 -7. The measure of an inscribed angle is INSCRIBED ANGLE THEOREM 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.
Thrm 9 -7. The measure of an inscribed angle is INSCRIBED ANGLE THEOREM 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 0 11 55 Z Intercepted Arc
Thrm 9 -7. Thethe measure of anofinscribed Find value x andangle y is equal to ½ 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 angles intercept Find value of 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 m PRQ = ½ m. PR
What is m PRQ ? 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 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. A D 1 B C
Interior Angle Theorem A 91 C y° x° 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. )
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
Exterior Angle Theorem • The measure of the angle formed is equal to ½ the difference of the intercepted arcs. k j 1 k j 3
PUTTING IT TOGETHER! D 6 C E A 3 Q 2 1 5 4 G F • • • AF is a diameter. m. AG=100 m. CE=30 m. EF=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 m PSR + m PQR = 180 S R
- -dot
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