Answer:
The angle bisectors of a triangle intersect at a point called the incenter. In this case, the angle bisectors drawn from vertices A and B intersect at point D.
To find the measure of angle ADB (m∠ADB), we need to use the fact that the angle bisectors of a triangle divide the opposite side into segments that are proportional to the adjacent sides.
Let's assume that the length of AD is x and the length of BD is y. Since the angle bisectors divide the opposite side in a ratio equal to the ratio of the adjacent sides, we can set up the following proportion:
AC/BC = AD/BD
Now, since AC is opposite to angle C and BC is opposite to angle B, we can write the equation as:
sin(γ)/sin(∠C) = x/y
Simplifying this equation, we get:
y * sin(γ) = x * sin(∠C)
Now, to find m∠ADB, we need to find the value of x/y. We can do this by solving for x and y using the properties of triangles.
Using the Law of Sines, we know that in a triangle:
sin(∠C)/AC = sin(∠B)/BC
Since AC is opposite to angle C and BC is opposite to angle B, we can rewrite the equation as:
sin(∠C)/AD = sin(∠B)/BD
Simplifying this equation, we get:
BD * sin(∠C) = AD * sin(∠B)
Now, we can substitute this equation into the previous equation we derived:
y * sin(γ) = (AD * sin(∠B))
Simplifying further:
y = (AD * sin(∠B))/sin(γ)
Now, we have the values of x/y and y. We can substitute these values into the original equation:
sin(γ)/sin(∠C) = x/y
Substituting the values of x/y and y, we get:
sin(γ)/sin(∠C) = x/((AD * sin(∠B))/sin(γ))
Simplifying this equation, we get:
sin(γ)^2/sin(∠C) = x/(AD * sin(∠B))
Finally, to find the measure of angle ADB (m∠ADB), we can use the inverse sine function:
m∠ADB = arcsin(x/(AD * sin(∠B)))
This formula allows us to find the measure of angle ADB (m∠ADB) based on the given angle C (γ) and the properties of the triangle.
Explanation: