Given:
Given that the difference of measures between the arcs subtended by chord AB is 160°.
Line l is tangent to the circle at point A.
We need to determine the measure of the angle between the tangent l and secant AB.
Measure of arc AB:
Let x denote the measure of minor arc AB.
Thus, we have;
[tex]x+(x+160^{\circ})=360^{\circ}[/tex]
[tex]2x+160^{\circ}=360^{\circ}[/tex]
[tex]2x=200^{\circ}[/tex]
[tex]x=100^{\circ}[/tex]
Thus, the measure of the minor arc AB is 100°
Measure of angle between the tangent l and the secant AB:
Since, we know the property that, "if a tangent and the chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc".
Thus, we have;
Measure of angle = [tex]\frac{1}{2} m( \widehat{AB})[/tex]
Substituting the value, we get;
Measure of angle = [tex]\frac{1}{2} (100^{\circ}) = 50^{\circ}[/tex]
Thus, the measure of angle between the tangent l and the secant AB is 50°
