To find the measure of angle ABC, we can use a property of tangents and circles. When two tangents are drawn from a point outside the circle, the angle formed between the two tangents equals half the difference of the measures of the arcs intercepted by the angle.
In our case, since AB and BC are tangents to the circle, and the measure of arc AC is given as 130 degrees, we first need to determine the measure of the arc opposite to angle ABC, which is arc AC. The complete circle is 360 degrees, so the measure of arc BC (the arc not included in angle ABC) can be calculated as follows:
Measure of arc BC = 360 degrees – Measure of arc AC = 360 degrees – 130 degrees = 230 degrees.
Now, we can find the measure of angle ABC using the formula:
Measure of angle ABC = 1/2 * (Measure of arc AC – Measure of arc BC) = 1/2 * (130 degrees – 230 degrees).
Calculating this gives:
Measure of angle ABC = 1/2 * (-100 degrees) = -50 degrees.
This result suggests that we should take the absolute value of the angle measurement since angles cannot be negative. Therefore, the measure of angle ABC is:
Measure of angle ABC = 50 degrees.
In conclusion, the measure of angle ABC is 50 degrees, and it results from the relationship between the angles and arcs created by tangents drawn from an external point to a circle.