Question

2. Угол при основании равнобедренного треугольника АВС равен 32º, АВ -его боковая сторона, АМ- биссектриса треугольника. Найдите углы треугольника АВМ. (Рассмотрите два случая.)

Answer 1

Все решение находится на картинке, но второго случая там нет. Извините уж.

Answer 2

Problem Analysis

We are given that triangle ABC is an isosceles triangle with angle BAC equal to 32º. AB is one of the equal sides, and AM is the angle bisector of triangle ABC. We need to find the angles of triangle ABM.

Solution

To find the angles of triangle ABM, we can use the properties of isosceles triangles and the angle bisector theorem.

Case 1: Angle BAC is the vertex angle of triangle ABC In this case, angle BAC is the largest angle in triangle ABC. Since AB is one of the equal sides, angle ABC and angle ACB are also equal. Let's denote angle ABC (or angle ACB) as x.

According to the angle bisector theorem, the angle bisector AM divides the side BC into two segments, BM and MC, such that the ratio of the lengths of BM and MC is equal to the ratio of the lengths of AB and AC. Since triangle ABC is isosceles, AB = AC, and the ratio of BM to MC is 1:1.

Therefore, angle ABM = angle MBC = x/2, and angle AMC = angle ACB = x.

Case 2: Angle BAC is one of the base angles of triangle ABC In this case, angle BAC is one of the smaller angles in triangle ABC. Since AB is one of the equal sides, angle ABC and angle ACB are also equal. Let's denote angle ABC (or angle ACB) as x.

According to the angle bisector theorem, the angle bisector AM divides the side BC into two segments, BM and MC, such that the ratio of the lengths of BM and MC is equal to the ratio of the lengths of AB and AC. Since triangle ABC is isosceles, AB = AC, and the ratio of BM to MC is 1:1.

Therefore, angle ABM = angle MBC = x/2, and angle AMC = angle ACB = x.

Summary

In both cases, the angles of triangle ABM are as follows: - Angle ABM = Angle MBC = x/2 - Angle AMC = Angle ACB = x

Please let me know if anything is unclear or if you have any further questions!