Question

Find the measure of MNQ

Answer 1

Answer:

m∠MNQ = 158

Step-by-step explanation:

As it can be seen in the figure:

+) The measure of arc MQ = 91 degree

+) The measure of arc RP = 225 degree

As this is the circle, four points M, Q, P and R are on the circle, so that we have:

+) m∠RMP = 1/2. measure of arc RP = 1/2 x 225 = 112.5 degree

As N is on MP

=> m∠RMN = m∠RMP = 112.5

+) m∠ MRQ = 1/2 measure of arc MQ = 1/2 x 91 = 45.5 degree

As N is on RQ

=> m∠MRN = m∠MRQ = 45.5

In the triangle RMN, the total measure of 3 internal angles is equal to 180 degree, so that:

m∠MNR + m∠RMN + m∠MRN = 180

=> m∠MNR + 112.5 + 45.5 = 180

=> m∠MNR  = 180 -112.5 -45.5 = 22

As N is on QR

=> m∠MNR + m∠MNQ = 180

=> m∠MNQ = 180 - m∠MNR  = 180 - 22 = 158

So that m∠MNQ = 158