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 $a$?

11 possible measures

Step-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°

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.

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step-by-step explanation:

step-by-step explanation:

two pairs of adjacent sides are of equal length. this implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. it also implies that the diagonals are perpendicular.