Question

В треугольнике авс проведена бисектриса ал ,алс равен 120 ,угол авс равен 100 .Найти угол асв .ответ дайте в градусах

Answer 1

1) угол ALC и угол ALB-смежные, следовательно угол ALC=180-120=60
2) угол ALB+угол ABC+угол BAL=180
60+100+угол BAL=180
угол BAL=180-60-100
угол BAL=20
3) угол BAL равен углу LAC(т.к AL-бис-са)
4) рассмотрим треугольник ALC:
угол LAC=20
угол LAC+угол ALC+угол ACB=180
20+120+угол ACB=180
угол ACB=180-120-20
угол ACB=60-20
угол ACB=40
ответ:40

Answer 2

Problem Analysis

We are given a triangle ABC, where the bisector AL is drawn, and angle ALS is equal to 120 degrees, and angle ABC is equal to 100 degrees. We need to find the measure of angle ACS.

Solution

To find the measure of angle ACS, 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 AL as x, and the length of segment LC as y. Then, we can write the following proportion:

AL / LC = AB / BC

Substituting the given values, we have:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can write:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Now, we can use the law of sines to find the lengths of AB and BC in terms of x and y. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In triangle ABC, we have:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values, we have:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further, we get:

AB / sin(angle ACS) = BC / sqrt(3)

Now, we can substitute the expression for AB / BC from the angle bisector theorem:

(x / y) / sin(angle ACS) = 1 / sqrt(3)

Simplifying further, we get:

x / (y * sin(angle ACS)) = 1 / sqrt(3)

Rearranging the equation, we have:

sin(angle ACS) = sqrt(3) * x / y

Finally, we can solve for angle ACS by taking the inverse sine of both sides:

angle ACS = arcsin(sqrt(3) * x / y)

Now, let's substitute the given values and calculate the angle ACS.

Calculation

Given: - Angle ALS = 120 degrees - Angle ABC = 100 degrees

Using the angle bisector theorem, we can find the ratio of AL to LC:

AL / LC = AB / BC

Substituting the given values:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can find angle BAC:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Using the law of sines, we can find the lengths of AB and BC in terms of x and y:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further:

AB / sin(angle ACS) = BC / sqrt(3)

Substituting the expression for AB / BC from the angle bisector theorem:

(x / y) / sin(angle ACS) = 1 / sqrt(3)

Rearranging the equation:

sin(angle ACS) = sqrt(3) * x / y

Taking the inverse sine of both sides:

angle ACS = arcsin(sqrt(3) * x / y)

Now, let's substitute the given values and calculate the angle ACS.

Calculation

Given: - Angle ALS = 120 degrees - Angle ABC = 100 degrees

Using the angle bisector theorem, we can find the ratio of AL to LC:

AL / LC = AB / BC

Substituting the given values:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can find angle BAC:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Using the law of sines, we can find the lengths of AB and BC in terms of x and y:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further:

AB / sin(angle ACS) = BC / sqrt(3)

Substituting the expression for AB / BC from the angle bisector theorem:

(x / y) / sin(angle ACS) = 1 / sqrt(3)

Rearranging the equation:

sin(angle ACS) = sqrt(3) * x / y

Taking the inverse sine of both sides:

angle ACS = arcsin(sqrt(3) * x / y)

Now, let's substitute the given values and calculate the angle ACS.

Calculation

Given: - Angle ALS = 120 degrees - Angle ABC = 100 degrees

Using the angle bisector theorem, we can find the ratio of AL to LC:

AL / LC = AB / BC

Substituting the given values:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can find angle BAC:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Using the law of sines, we can find the lengths of AB and BC in terms of x and y:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further:

AB / sin(angle ACS) = BC / sqrt(3)

Substituting the expression for AB / BC from the angle bisector theorem:

(x / y) / sin(angle ACS) = 1 / sqrt(3)

Rearranging the equation:

sin(angle ACS) = sqrt(3) * x / y

Taking the inverse sine of both sides:

angle ACS = arcsin(sqrt(3) * x / y)

Now, let's substitute the given values and calculate the angle ACS.

Calculation

Given: - Angle ALS = 120 degrees - Angle ABC = 100 degrees

Using the angle bisector theorem, we can find the ratio of AL to LC:

AL / LC = AB / BC

Substituting the given values:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can find angle BAC:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Using the law of sines, we can find the lengths of AB and BC in terms of x and y:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further:

AB / sin(angle ACS) = BC / sqrt(3)

Substituting the expression for AB / BC from the angle bisector theorem:

(x / y) / sin(angle ACS) = 1 / sqrt(3)

Rearranging the equation:

sin(angle ACS) = sqrt(3) * x / y

Taking the inverse sine of both sides:

angle ACS = arcsin(sqrt(3) * x / y)

Now, let's substitute the given values and calculate the angle ACS.

Calculation

Given: - Angle ALS = 120 degrees - Angle ABC = 100 degrees

Using the angle bisector theorem, we can find the ratio of AL to LC:

AL / LC = AB / BC

Substituting the given values:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can find angle BAC:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Using the law of sines, we can find the lengths of AB and BC in terms of x and y:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further:

AB / sin(angle ACS) = BC / sqrt(3)

Substituting the expression for AB / BC from the angle bisector theorem:

(x / y) / sin(angle ACS) = 1 / sqrt(3)

Rearranging the equation:

sin(angle ACS) = sqrt(3) * x / y

Taking the inverse sine of both sides:

angle ACS = arcsin(sqrt(3) * x / y)

Now, let's substitute the given values and calculate the angle ACS.

Calculation

Given: - Angle ALS = 120 degrees - Angle ABC = 100 degrees

Using the angle bisector theorem, we can find the ratio of AL to LC:

AL / LC = AB / BC

Substituting the given values:

x / y = AB / BC

Since angle ALS is equal to 120 degrees, we can find angle BAC:

angle BAC = 180 - angle ALS = 180 - 120 = 60 degrees

Using the law of sines, we can find the lengths of AB and BC in terms of x and y:

AB / sin(angle ACS) = BC / sin(angle BAC)

Substituting the known values:

AB / sin(angle ACS) = BC / sin(60)

Simplifying further:

AB / sin(angle ACS) = BC / sqrt(3)

Substituting the expression for AB / BC from the angle bisector theorem:

**(x / y) / sin(angle ACS) = 1