Question

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 1

11 possible measures

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°

`\implies m\angle B=(90)/(x+1)`

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 90

1 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 90

i.e. there are 11 possible values of m∠B.

Hence, 11 measures are possible for angle A.