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69e.) given: measure of arc iv=114°, m∠vsk=38° find: measure of arc pk

69d.) given: measure of arc piv=7/2 times measure of arc pkv find: m∠vpj

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69e.) 170

69d.) 140

Step-by-step explanation:

69e.)

This is a case of two secants that intersect inside a circle.

Segments IK and PV are secants that intersect inside circle O.

m<PSK = (1/2)(m(arc)PK + m(arc)IV)

m<VSK + m<PSK = 180

m<VSK = 38

m<PSK = 180 - 38

m<PSK = 142

142 = (1/2)(m(arc)PK + 114)

284 = m(arc)PK + 114

m(ark)PK = 170

69d.)

Let m(arc)PKV = x

Then m(arc)PIV = 7/2(m(arc)PKV) = 3.5x

A full circle has an arc angle measure of 360.

x + 3.5x = 360

4.5x = 360

x = 80

m(arc)PKV = 80

The measure of an angle with vertex on a circle formed by the intersection of a tangent and a chord is half the measure of the intercepted arc.

m<VPL = (1/2)m(arc)PKV

m<VPL = (1/2)(80)

m<VPL = 40

Angles VPL and VPJ are supplementary, so the sum of their measures equals 180 deg.

m<VPL + m<VPJ = 180

m<VPJ = 180 - m<VPL

m<VPJ = 180 - 40

m<VPJ = 140

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step-by-step explanation: