Angles between the Tangent and the Chord

Here we will prove that if a line touches a circle and from the point of contact a chord is down, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.

Given: A circle with centre O. Tangent XY touches the circle at the point M. Through M, a chord MN is drawn. Let MN subtend ∠MSN and ∠MTN in the major and minor segments respectively.

To prove: ∠NMY = ∠MSN and ∠NMX = ∠MTN

Construction: Draw the diameter MOR. Join N to R.

Proof:

 Statement: Reason 1. ∠RMY = 90°⟹ ∠RMN + ∠NMY = 90°⟹ ∠NMY = 90° - ∠RMN 1. Diameter ⊥ Tangent. 2. In ∆RMN, ∠MNR = 90° 2. Angle in a semicircle is 90°. 3. ∠NRM + ∠RMN = 90° 3. In a right-angled triangle, sum of the two acute angles is 90°. 4. ∠NRM = ∠MSN 4. Angles in the same segment are equal. 5. ∠MSN + ∠RMN = 90°⟹ ∠MSN = 90° - ∠RMN 5. From statements 3 and 4. 6. ∠NMY = ∠MSN 6. From statements 1 and 5. 7. ∠NMY + ∠NMX = 180° 7. Linear pair. 8.  ∠MSN + ∠MTN = 180° 8. Opposite angles of a cyclic quadrilateral are supplementary. 9. ∠NMY + ∠NMX = ∠MSN + ∠MTN 9. From 7 and 8. 10. ∠NMX = ∠MTN. 10. ∠NMY = ∠MSN from statement 6.

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