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Tangent and Intersected Chord Theorem

Rule

Tangent and Intersected Chord Theorem

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.

Circle, tangent, chord, two angles and two arcs

Based on the diagram, the following relation holds.

and

Keep in mind that this theorem is true also in the case that is a secant.

Proof

Proof

Consider a diameter Since is tangent to the circle at then

Circle, tangent, chord, diamter, two angles and two arcs

In addition, since is a diameter, is a semicircle and then its measure is The Arc Addition Postulate allows setting the following equation. Since is a central angle, its measure is half the measure of the intercepted arc. That is, or equivalently, Similarly, the Angle Addition Postulate can be used to establish the following equation. Next, multiply the equation above by and add it to the previous equation. Solving the last equation by gives the first desired equation.

Once more, the Arc Addition Postulate and the Angle Addition Postulate can be applied to set the following pair of equations. Recall that and Next, multiply Equation by and add it to Equation Finally, by solving the resulting equation for the second desired equation is obtained.