Question

В треугольнике ABC A = 43°, C = 59°. Через вершину В проведена прямая MN || АС. Найдите угол MBD, где BD - биссектриса угла ABC Помогите пожалуйста!!!​

Answer 1

Ответ:

82 и 98. их два нажми добавить поле.

Answer 2

Problem Analysis

We are given a triangle ABC with angle A measuring 43° and angle C measuring 59°. A line MN is drawn through vertex B, parallel to AC. We need to find the measure of angle MBD, where BD is the angle bisector of angle ABC.

Solution

To find the measure of angle MBD, we need to determine the measure of angle ABC first. We can do this by using the fact that the sum of the angles in a triangle is always 180°.

Let's calculate the measure of angle ABC: - Angle A = 43° - Angle C = 59°

Using the fact that the sum of the angles in a triangle is 180°, we can calculate angle B: - Angle B = 180° - Angle A - Angle C - Angle B = 180° - 43° - 59° - Angle B = 78°

Now that we know the measure of angle ABC is 78°, we can find the measure of angle MBD. Since BD is the angle bisector of angle ABC, we can use the angle bisector theorem to find the measure of angle MBD.

According to the angle bisector theorem, the ratio of the lengths of the sides of the triangle formed by the angle bisector is equal to the ratio of the lengths of the other two sides of the original triangle.

Let's denote the length of side AB as a, the length of side BC as b, and the length of side AC as c. Then, according to the angle bisector theorem:

BD/DC = AB/AC

Since MN is parallel to AC, we can use the alternate interior angles theorem to conclude that angle MBD is congruent to angle ABC. Therefore, we can rewrite the angle bisector theorem equation as:

BD/DC = AB/BC

Now, let's substitute the values we know into the equation: - AB = a - BC = b - AC = c

We can rewrite the equation as:

BD/DC = a/b

Since BD is the angle bisector, we know that BD/DC = AB/AC. Therefore, we can rewrite the equation as:

AB/AC = a/b

Substituting the known values, we get:

a/c = a/b

Cross-multiplying, we get:

ab = ac

Dividing both sides by a, we get:

b = c

Since angle B is congruent to angle MBD, we can conclude that angle MBD is also 78°.

Answer

Therefore, the measure of angle MBD is 78°.

Please let me know if you need any further clarification or assistance!