4). Triangle ABC. Angle B is 9 times smaller than Angle A. Find Angle B and Angle A.

Solution:

1. Let angle A be represented by 'x'.
2. According to the problem, angle B is 9 times smaller than angle A, so angle B = x / 9.
3. The triangle ABC is a right triangle with the right angle at C (indicated by the square symbol). Therefore, angle C = 90 degrees.
4. The sum of angles in a triangle is 180 degrees.
5. So, angle A + angle B + angle C = 180 degrees.
6. Substitute the expressions for angles A and B, and the value of angle C into the equation: x + (x / 9) + 90 = 180.
7. Subtract 90 from both sides: x + (x / 9) = 180 - 90.
8. x + (x / 9) = 90.
9. To add x and x/9, find a common denominator, which is 9: (9x / 9) + (x / 9) = 90.
10. Combine the terms: (9x + x) / 9 = 90.
11. 10x / 9 = 90.
12. To solve for x, multiply both sides by 9: 10x = 90 * 9.
13. 10x = 810.
14. Divide both sides by 10: x = 810 / 10.
15. x = 81. So, angle A = 81 degrees.
16. Now, find angle B: angle B = x / 9 = 81 / 9 = 9 degrees.
17. Let's check if the sum of angles is 180: Angle A + Angle B + Angle C = 81° + 9° + 90° = 180°. The sum is correct.

Ответ: ∠B = 9°, ∠A = 81°