Question

Треугольник АВС-равнобедренный, АС-основание, угол В=64°, СМ-биссектриса Найти угол АМС?

Answer 1

Тут надо было рассматривать то, что треугольник равнобедренный и бессикриса делит угол пополам

Answer 2

Task: Finding angle AMS in an isosceles triangle ABC

To find the angle AMS in the isosceles triangle ABC, where AC is the base and angle B is 64°, we can use the properties of isosceles triangles and the angle bisector theorem.

Solution:

1. In an isosceles triangle, the base angles are equal. Therefore, angle A = angle C.

2. Let's denote angle A as x. So, angle C is also x.

3. Since angle B is given as 64°, we can find angle A by subtracting 64° from 180° (the sum of the angles in a triangle). So, angle A = 180° - 64° = 116°.

4. Now, we know that angle A = angle C = 116°.

5. The angle bisector of angle A divides the opposite side (BC) into two segments in the ratio of the adjacent sides (AB and AC). Let's denote the point where the angle bisector intersects BC as M.

6. According to the angle bisector theorem, the ratio of BM to MC is equal to the ratio of AB to AC.

7. Since triangle ABC is isosceles, AB = AC. Therefore, the ratio of BM to MC is 1:1.

8. This means that BM = MC.

9. Since BM = MC, triangle BMC is an isosceles triangle.

10. In an isosceles triangle, the base angles are equal. Therefore, angle BMC = angle MBC.

11. Let's denote angle BMC as y. So, angle MBC is also y.

12. Since angle BMC = angle MBC, we can write the equation: 2y = 180° - x.

13. Solving for y, we have: y = (180° - x) / 2.

14. Substituting the value of x (116°) into the equation, we get: y = (180° - 116°) / 2 = 64° / 2 = 32°.

15. Therefore, angle BMC = angle MBC = 32°.

16. Finally, to find angle AMS, we subtract angle BMC from angle B: AMS = 64° - 32° = 32°.

Answer:

The angle AMS in the given isosceles triangle ABC, where AC is the base and angle B is 64°, is 32°.