Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
Equal angles.
Step 1: Draw a circle with center \(O\). Take a chord \(PQ\) of this circle.
Step 2: At point \(P\), draw a tangent to the circle. Similarly, at point \(Q\), draw a tangent to the circle.
Step 3: Let the angle made by the tangent at \(P\) with the chord \(PQ\) be \(\alpha\). Let the angle made by the tangent at \(Q\) with the chord \(PQ\) be \(\beta\).
Step 4: Recall the Tangent–Chord Theorem: "The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment of the circle."
Step 5: Apply this theorem at point \(P\): The angle between the tangent at \(P\) and the chord \(PQ\) (i.e., \(\alpha\)) is equal to the angle formed at the opposite arc (say at point \(R\) on arc \(Q\widehat{P}\)).
Step 6: Apply the theorem at point \(Q\): The angle between the tangent at \(Q\) and chord \(PQ\) (i.e., \(\beta\)) is equal to the angle formed at the opposite arc (say at the same point \(R\) on arc \(P\widehat{Q}\)).
Step 7: Both \(\alpha\) and \(\beta\) are equal to the same angle at point \(R\). Therefore, \(\alpha = \beta\).
Final Result: The tangents drawn at the ends of a chord of a circle make equal angles with the chord.