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# geometry

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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?

May 11, 2024

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Since angles A and B are complementary, their measures sum up to 90 degrees. Let's denote the measure of angle B as $$x$$ degrees. Then, the measure of angle A is $$90 - x$$ degrees.

Given that angle A is a multiple of angle B, we can express angle A as $$kx$$, where $$k$$ is a positive integer.

So, we have:

$kx = 90 - x$

Solving for $$x$$, we get:

$kx + x = 90$

$x(k + 1) = 90$

$x = \frac{90}{k + 1}$

Now, since $$x$$ must be a positive integer and a divisor of 90, let's list down the possible values of $$x$$ and then find corresponding values of $$k$$:

1. If $$k + 1 = 1$$, then $$x = 90$$, but $$x$$ cannot be 90 as it's an integer less than 90.

2. If $$k + 1 = 2$$, then $$x = 45$$, which satisfies the conditions.

3. If $$k + 1 = 3$$, then $$x = 30$$, which also satisfies the conditions.

4. If $$k + 1 = 5$$, then $$x = 18$$, which satisfies the conditions.

5. If $$k + 1 = 9$$, then $$x = 10$$, which satisfies the conditions.

6. If $$k + 1 = 10$$, then $$x = 9$$, but 9 is not a divisor of 90.

7. If $$k + 1 = 15$$, then $$x = 6$$, which satisfies the conditions.

8. If $$k + 1 = 18$$, then $$x = 5$$, which satisfies the conditions.

9. If $$k + 1 = 30$$, then $$x = 3$$, which satisfies the conditions.

10. If $$k + 1 = 45$$, then $$x = 2$$, which satisfies the conditions.

11. If $$k + 1 = 90$$, then $$x = 1$$, but 1 is not a divisor of 90.

Therefore, the possible measures of angle A, expressed as multiples of angle B, are 2, 3, 5, 9, 15, 18, 30, 45. So, there are 8 possible measures for angle A.

May 11, 2024