By the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc.
m∠MHL = 12m LH & (I) m∠GLH = 12m GH & (II)
Notice that ∠MHL is an exterior angle of △ FHL. Then, applying the Triangle Exterior Angle Theorem, we conclude that its measure is equal to the sum of the measures of the two nonadjacent interior angles.
m∠MHL = m∠GLH + m∠F
⇓
m∠F = m∠MHL - m∠GLH
Finally, we substitute the two equations found above into the last equation. That way we get the desired equation.
m∠F = 1/2m LH - 1/2m GH
⇓
m∠F = 1/2(m LH - m GH)
Two-Column Proof
Given: & TangentFMand secantFL
Prove: & m∠F = 12(mLH - mGH)
Let's summarize the proof we did above in the following two-column proof table.
