In the diagram below of triangle L, M, NLMN, OO is the midpoint of start overline, L, N, end overline LN and PP is the midpoint of start overline, M, N, end overline

MN . If mangle, N, L, M, equals, minus, 6, x, plus, 78∠NLM=−6x+78, and mangle, N, O, P, equals, 9, x, plus, 33∠NOP=9x+33, what is the measure of angle, N, O, P∠NOP?

The measure of angle NOP is determined by first finding the value of x from the given equations and then substitifying it back to get the measure of the angle. The feasible value of x is found to be 23, so NOP equals 240 degrees.

Explanation:

Since O is the midpoint of LN and P is the midpoint of MN, it forms straight angles which are equal to 180 degrees. That is, ∠NLM + ∠NLO = 180 and ∠NOP + ∠NPM = 180. Given that ∠NLM = -6x + 78 and ∠NOP = 9x + 33, you express NOP and NLM as a straight line (180 degrees). Therefore, -6x + 78 + 9x + 33 = 180. Simplifying the equation gives 3x + 111 = 180. By moving 111 to the other side, you get 3x = 180 - 111 = 69. Therefore, x = 69/3 = 23. Substituting x = 23 into ∠NOP = 9x + 33, we find that ∠NOP = 9*23 + 33 = 240 degrees.

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