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.

Angles between the Tangent and the Chord

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.





10th Grade Math

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