Angles in Circles Central Angles A central angle

  • Slides: 28
Download presentation
Angles in Circles

Angles in Circles

Central Angles • A central angle is an angle whose vertex is the CENTER

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 180 R

An angle inscribed in a semicircle is a right angle. P S 180 90

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

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

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

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

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

Exterior Angles • An angle formed by two secants, two tangents, or a secant

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

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

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

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

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