In the diagram, triangle MNR is inscribed in circle MNR and line PQ is a straight line. ∠MRN = 41 and = 141, find ∠QNR
Answer Details
We can use the property that an angle inscribed in a circle is half the measure of the arc it intercepts. Since triangle MNR is inscribed in circle MNR, we know that angle MNR is half the measure of arc MR. Similarly, angle MRN is half the measure of arc MN. We also know that arc PQM and arc QNR add up to a full circle, which has a measure of 360 degrees. Using these facts, we can write two equations: - angle MNR = 1/2(arc MR) = 1/2(360 - arc PQM - arc QNR) - angle MRN = 1/2(arc MN) = 1/2(360 - arc PQM) Substituting the given angle measures, we have: - 41 = 180 - arc PQM/2 - arc QNR/2 - 141 = 180 - arc PQM/2 Solving for arc PQM in the second equation gives us arc PQM = 198. Substituting this into the first equation, we have: 41 = 180 - 99 - arc QNR/2 arc QNR/2 = 40 arc QNR = 80 Therefore, angle QNR is half the measure of arc QNR, which is 80 degrees. So, the answer is 80º.