The measure of arcMN, arcNP and arcNQP is 65,115 and 245 degrees respectively
Circle theorem
From the given circle theorem, the measure of the arc MN is equal to PQ.
Given the following parameters
MN = 10x - 45
PQ = 6x - 1
Equate
10x - 45 = 6x - 1
10x -6x = -1 + 45
4x = 44
x = 11
Determine the measure of MN
MN = 10x - 45
MN = 10(11) - 45
MN = 110 - 45
MN = 65 degrees
arcNP = 180 - arcMN
arcNP = 180 - 65
arcNP = 115 degrees
arcNQP = 180 + arcPQ
arcNQP = 180 + 65
arcNQP = 245 degrees
Hence the measure of arcMN, arcNP and arcNQP is 65,115 and 245 degrees respectively
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