We haven been told that JL and NL are secant lines intersecting at point L.
We want to prove that m∠ JPN > m∠ JLN. Looking at the diagram, we can see that two chords intersect inside the circle and two secants intersect outside the circle. In this case, we will first recall the Angles Inside the Circle Theorem (Theorem 10.15) and Angles Outside the Circle Theorem (Theorem 10.16).
Angles Inside the Circle Theorem
If two chords intersect inside a circle, then the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Angles Outside the Circle Theorem
If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.
Notice that the intercepted arcs of both ∠ JPN and ∠ JLN are KM and JN.
Therefore, we can write the measures of ∠ JPN and ∠ JLN in terms of mKM and mJN.
m∠ JPN = 1/2(mJN+mKM)
m∠ JLN = 1/2(mJN-mKM)
From here we can immediately conclude that the sum of the intercepted arcs is greater than the difference of the intercepted arc. Consequently, m∠ JPN is greater than m∠ JLN.
