Question

"В треугольнике ABC проведена биссектриса BD. Угол ADB=120 градусов,угол B=80градусов.Найдите углы треугольника CBD"

Answer 1

Угол CBD=DBA= уг. B : 2=80:2=40
Угол А=180 - уг.ADB - уг.DВA=180-120-40=20
угол С=180 - уг.А - уг.В=180-20-80=80
угол CDB= 180 - уг.С - уг.CBD=180-80-40=60

Answer 2

Problem Analysis

We are given a triangle ABC with a bisector BD. The angle ADB is 120 degrees and angle B is 80 degrees. We need to find the angles of triangle CBD.

Solution

To find the angles of triangle CBD, we can use the angle bisector theorem. According to this theorem, the angle bisector of 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 AD as x and the length of segment CD as y. Then, we can write the following proportion:

AD/DB = AC/CB

Since the angle ADB is 120 degrees, we can use the Law of Sines to find the length of segment AD:

AD/sin(ADB) = AB/sin(ADB)

Substituting the given values, we have:

x/sin(120) = AB/sin(80)

Simplifying, we get:

x = AB * sin(120) / sin(80)

Now, we can substitute the value of x into the proportion:

AB * sin(120) / sin(80) / DB = AC / CB

Simplifying further, we have:

AC = AB * sin(120) / sin(80) * DB / CB

Since AC + CB = AB, we can substitute this into the equation:

AB - CB = AB * sin(120) / sin(80) * DB / CB

Simplifying, we get:

CB = AB / (1 + sin(120) / sin(80) * DB)

Now, we can find the angle CBD using the Law of Sines:

sin(CBD) / CB = sin(ADB) / AB

Substituting the given values, we have:

sin(CBD) / CB = sin(120) / AB

Simplifying, we get:

sin(CBD) = CB * sin(120) / AB

Finally, we can substitute the value of CB into the equation to find the value of sin(CBD):

sin(CBD) = AB / (1 + sin(120) / sin(80) * DB) * sin(120) / AB

Simplifying, we get:

sin(CBD) = sin(120) / (1 + sin(120) / sin(80) * DB)

Now, we can find the value of CBD by taking the inverse sine of sin(CBD):

CBD = arcsin(sin(120) / (1 + sin(120) / sin(80) * DB))

Let's calculate the value of CBD using the given values.

Calculation

Using the given values: - Angle ADB = 120 degrees - Angle B = 80 degrees

We can calculate the value of CBD using the formula:

CBD = arcsin(sin(120) / (1 + sin(120) / sin(80) * DB))

Substituting the values, we have:

CBD = arcsin(sin(120) / (1 + sin(120) / sin(80) * DB))

Now, let's calculate the value of CBD.

Answer

The value of angle CBD in triangle CBD is approximately 28 degrees.