The measures of angles $A$ and $B$ are both positive, integer numbers of degrees. The measure of angle $A$ is a multiple of the measure of angle $B$, and angles $A$ and $B$ are complementary angles. How many measures are possible for angle $A$?

Answer:11 possible measuresStep-by-step explanation:Given,Measures of angles ∠A and ∠B are positive integer numbers degree.Such that,Measure of angle A is a multiple of the measure of angle B,That is,m∠A = x(m∠B)Where, x is any positive number.If angle A and angle B are complementary angles,Then m∠A + m∠B = 90°⇒ x(m∠B)  + m∠B = 90°⇒ (x+1) m∠B = 90°[tex]\implies m\angle B=\frac{90}{x+1}[/tex]Since, m∠B will be positive integer.If x + 1 = a factor of 90,∵ Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 901 can not possible ( because m∠A + m∠B = 90° )Thus, the possible values of x + 1 are,2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90i.e. there are 11 possible values of m∠B.Hence, 11 measures are possible for angle A.