Question

given circle i with chords \\( \overline { j h } \\), \\( \overline { j l } \\), \\( \overline { k h } \\), and \\( \overline { k l } \\), and \\( \overleftrightarrow { m j n } \\) tangent to i at point j. find an angle that is congruent to \\( \angle j l k \\).

Explanation:

Step1: Use the tangent - chord angle theorem

The measure of an angle formed by a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord.
For tangent \(\overleftrightarrow{MJN}\) and chord \(JL\), \(\angle NJL\) is formed.

Step2: Use the property of inscribed angles subtended by the same arc

\(\angle JHK\) and \(\angle JLK\) are inscribed angles subtended by arc \(JK\). Also, \(\angle NJL\) (angle between tangent \(MJN\) and chord \(JL\)) and \(\angle JHK\) are related by the tangent - chord angle theorem. In fact, \(\angle NJL=\angle JHK\) (tangent - chord angle theorem) and \(\angle JHK = \angle JLK\) (inscribed angles subtended by the same arc \(JK\)). Another way:
By the tangent - chord angle theorem, \(\angle MJH\) (angle between tangent \(MJN\) and chord \(JH\)) is equal to the inscribed angle subtended by arc \(JH\).
We know that \(\angle JLK\) and \(\angle MJH\) are equal. The tangent - chord angle theorem states that \(\angle MJH\) (angle between tangent \(MJN\) and chord \(JH\)) is congruent to the inscribed angle subtended by the arc \(JH\) that is not adjacent to \(\angle MJH\). And \(\angle JLK\) is an inscribed angle subtended by arc \(JH\) (since in a circle, inscribed angles subtended by the same arc are equal).

Answer:

\(\angle MJH\)