By the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc.
m∠VTS = 12m SWT & (I) m∠STR = 12m ST & (II)
Notice that ∠VTS is an exterior angle of △ STR. 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∠VTS = m∠RST + m∠R
⇓
m∠R = m∠VTS - m∠RST
Finally, we substitute the two equations found above into the last equation. That way we get the desired equation.
m∠R = 1/2m SWT - 1/2m RV
⇓
m∠R = 1/2(m SWT - m RV)
Two-Column Proof
Given: & TangentsRSand RV
Prove: & m∠R = 12(mSWT - mRV)
Let's summarize the proof we did above in the following two-column proof table.
