To find the measure of angle ABC, we first need to understand how the angle relates to the arc in a circle. The measure of angle ABC is half the measure of the intercepted arc AC.
We know the measure of arc AC is given as:
Arc AC = 8x + 8
According to the inscribed angle theorem, we can express angle ABC in terms of arc AC:
Angle ABC = 1/2 (Arc AC)
Substituting the value of arc AC into the equation:
Angle ABC = 1/2 (8x + 8)
Angle ABC = 4x + 4
Now, we also have the measure of angle ADC provided as:
Angle ADC = 7x + 2
Since angle ADC and angle ABC are related as both are inscribed angles that intercept the same arc AC, we can derive a relationship between them. This means:
Angle ADC = Angle ABC
Now, we set the two expressions equal to each other:
7x + 2 = 4x + 4
To solve for x, we simplify the equation:
7x – 4x = 4 – 2
3x = 2
x = 2/3
Now that we have the value of x, we can substitute it back into the expression for angle ABC:
Angle ABC = 4x + 4 = 4(2/3) + 4
Angle ABC = (8/3) + 4 = (8/3) + (12/3) = 20/3
This means that:
Angle ABC = 20/3 degrees
Hence, the measure of angle ABC is:
Angle ABC = 20/3 degrees