∠1 and ∠2 are a linear pair. m∠1 = x - 29, and m∠2 = x + 61. find the measure of each angle.

Since the angles are a linear pair, then we have to validate the following equation:
 m∠1 + m∠2 = 180
 Substituting the values we have:
 (x - 29) + (x + 61) = 180
 We clear x:
 2x + 32 = 180
 x = (180-32) / (2)
 x = 74
 Therefore, each angle will be:
 m∠1 = x - 29 = 74 - 29 = 45
 m∠2 = x + 61 = 74 + 61 = 135
 
 the measure of each angle are:
 m∠1 = 45
 m∠2 = 135