1. **State the problem:** Find the measure of angle $\angle KNM$ formed by a tangent and a secant intersecting outside the circle at point $K$. Given that $m\angle KNM = 151^\circ$ is the angle formed. 2. **Formula used:** The measure of an angle formed by a tangent and a secant intersecting outside a circle is half the difference of the measures of the intercepted arcs. Mathematically, $$m\angle = \frac{1}{2} |m\text{(major arc)} - m\text{(minor arc)}|$$ 3. **Explanation:** The angle outside the circle is half the difference of the intercepted arcs on the circle. 4. **Apply the formula:** Here, $m\angle KNM = 151^\circ$ is given, so $$151 = \frac{1}{2} |m(\widehat{KNM}) - m(\widehat{JL})|$$ 5. **Solve for the difference of arcs:** Multiply both sides by 2: $$2 \times 151 = |m(\widehat{KNM}) - m(\widehat{JL})|$$ $$302 = |m(\widehat{KNM}) - m(\widehat{JL})|$$ 6. **Interpretation:** The difference between the measures of arcs $\widehat{KNM}$ and $\widehat{JL}$ is $302^\circ$. Since the total circle is $360^\circ$, this means one arc is $302^\circ$ larger than the other. **Final answer:** The measure of $\angle KNM$ is $151^\circ$, which equals half the difference of the intercepted arcs $\widehat{KNM}$ and $\widehat{JL}$.