An electric winch is used to pull a boat out of the water onto a trailer.  The winch winds the cable around a circular drum of diameter 5 inches.  Approximately how many times will the winch have to rotate in order to roll in 5 feet of cable?

Solve the system of equations

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19y + 22x =180

40x +10y =180

⎧ ⎨ ⎩

x=3, y=6

Warm-upUse a compass to draw a circle

and label the center C.Draw an inscribed angle and

label it ADB.Draw .1. Measure ∠ADB and ∠ACB2. Find the mXZ and compare it

with m∠ADB.3. Make a conjecture.

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WX and WZ

10.3 Inscribed Angles

•Find measures of inscribed angles•Find measures of angles of inscribed polygons

An inscribed angle An angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

inscribed angle

intercepted arc

Theorem 10.8 Measure of an Inscribed AngleIf an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

m∠ADB= ½ mAB

Theorem 10.9If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

∠C≅∠D

Find the measure of the arc or angle.1. mADC 2. mAC

3. m∠ABC

14001800

980

Find the m∠ABE, m∠ACE, and m∠ADE.

450, 450, 450

Theorem 10.10 If a right triangle is inscribed in

a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

∠B is a right angle iff €

AC is a diameter of the circle.

Theorem 10.11 A quadrilateral can be

inscribed in a circle iff its opposite angles are supplementary. D, E, F, and G lie on some

circle, ⨀C, iff m∠D+m∠F = 1800 and m∠E +m∠G = 1800.

Find the value of y and x.

x = 95, y = 100

mA∠=600, m∠B=1140, m∠C=1200, m∠D=660

In the diagram, ABCD is inscribed in ⨀P. Find the measure of each angle.