Ray Np is an angle bisector of angle MNQ and measure of Angle PNQ = 2x+1. Find the measure of angle MNQ. If The measure of MNQ=x^2-10. Find the value of x, and the appropriate measure of angle MNQ

Refer to the diagram below.

Because ray NP bisects ∠MNQ, therefore

∠MNP = ∠PNQ = 2x + 1.

Therefore

∠MNQ = 2*∠PNQ = 2 (2x + 1) = 4x + 2.

Because ∠MNQ is given as x² - 10, therefore

x² - 10 = 4x + 2

x² - 4x - 12 = 0

(x + 2) (x - 6) = 0

x = - 2, or x = 6

When x = - 2,

∠MNQ = 4 * (-2) + 2 = - 6°

This answer is not acceptablle, therefore x = - 2 should be rejected.

When x = 6,

∠MNQ = 4*6 + 2 = 26°

Answer: x = 6, and ∠MNQ = 26°