Question

В треугольнике АВС угол А=100°. Биссектрисы СР и ВМ пересекаются в точке О. Найдите угол ВОС. Напишите, пожалуйста, ход решения задачи, а не просто ответ - ЭТО ВАЖНО!

Answer 1

Угол В+угол С=180⁰- уголА=180⁰-100⁰=80⁰
Так как СР и ВМ- биссектрисы , то 
угол ОВС+угол ОСВ=80⁰:2=40⁰
угол ОВС+угол ОСВ+угол ВОС=180 ⁰( в ΔВОС)
Значит угол ВОС=180⁰-(угол ОВС+угол ОСВ)=180⁰-40⁰=140⁰
Ответ:140⁰

Answer 2

Problem Analysis

We are given a triangle ABC with angle A measuring 100°. The bisectors of angles C and B intersect at point O. We need to find the measure of angle VOC.

Solution

To find the measure of angle VOC, we can use the angle bisector theorem. According to this theorem, the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides.

Let's denote the length of segment AO as x and the length of segment CO as y. Since the bisectors of angles C and B intersect at point O, we can apply the angle bisector theorem to triangle ABC to find the ratio of the lengths of the sides.

According to the angle bisector theorem, we have:

AC / AB = CO / OB

Since angle A measures 100°, angle B measures (180° - 100°) = 80°. Therefore, angle COB measures half of angle B, which is 40°.

Now, we can use the angle bisector theorem to write the equation:

AC / AB = CO / OB

Substituting the values we know:

AC / AB = y / x

We can rewrite this equation as:

AC * x = AB * y

Now, let's consider triangle AOC. The sum of the angles in a triangle is 180°, so we have:

angle AOC + angle OAC + angle OCA = 180°

Since angle AOC is the sum of angles A and COA, we have:

angle AOC = angle A + angle COA = 100° + angle COA

Similarly, angle OAC is the sum of angles OCA and COA, so we have:

angle OAC = angle OCA + angle COA = angle OCA + 40°

Substituting these values into the equation for the sum of angles in triangle AOC, we get:

100° + angle COA + angle OCA + 40° + angle OCA = 180°

Simplifying the equation, we have:

140° + 2 * angle OCA + angle COA = 180°

Now, let's consider triangle COB. The sum of the angles in a triangle is 180°, so we have:

angle COB + angle CBO + angle BCO = 180°

Since angle COB is 40°, angle CBO is the sum of angles C and COB, so we have:

angle CBO = angle C + angle COB = angle C + 40°

Similarly, angle BCO is the sum of angles COB and B, so we have:

angle BCO = angle COB + angle B = 40° + 80° = 120°

Substituting these values into the equation for the sum of angles in triangle COB, we get:

40° + angle C + 120° + angle C = 180°

Simplifying the equation, we have:

2 * angle C + 160° = 180°

Now, we can solve these two equations simultaneously to find the values of angle OCA and angle COA.

From the equation 140° + 2 * angle OCA + angle COA = 180°, we can isolate angle COA:

angle COA = 180° - 140° - 2 * angle OCA

Substituting this value into the equation 2 * angle C + 160° = 180°, we get:

2 * angle C + 160° = 180° - 140° - 2 * angle OCA

Simplifying the equation, we have:

2 * angle C + 160° = 40° - 2 * angle OCA

Rearranging the equation, we have:

2 * angle C + 2 * angle OCA = 40° - 160°

Simplifying the equation, we have:

2 * angle C + 2 * angle OCA = -120°

Dividing both sides of the equation by 2, we get:

angle C + angle OCA = -60°

Now, we can substitute this value into the equation 2 * angle C + 2 * angle OCA = -120° to find the value of angle C:

angle C + angle OCA = -60°

angle C + (-60°) = -120°

Simplifying the equation, we have:

angle C - 60° = -120°

Adding 60° to both sides of the equation, we get:

angle C = -120° + 60°

Simplifying the equation, we have:

angle C = -60°

Since angle C cannot be negative, we can conclude that there is no valid solution for this problem.

Therefore, we cannot find the measure of angle VOC in this triangle.