Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

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In the study of circle geometry, one of the intriguing aspects is the relationship between tangents drawn from an external point and the angles these tangents form. A fascinating theorem in this context states that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center of the circle. This theorem not only enriches our understanding of circle geometry but also demonstrates the elegant interplay between tangents, angles, and radii.

A tangent to a circle is a line that touches the circle at exactly one point, known as the point of tangency. When two tangents are drawn from an external point to a circle, they touch the circle at two distinct points. These tangents, despite their individual straight paths, form an angle at the point from where they are drawn. This angle is crucial to our theorem and is the focus of our proof.

In our scenario, when we draw radii from the center of the circle to the points of tangency, we create two right angles where the radii meet the tangents. This is because a radius of a circle is always perpendicular to the tangent at the point of tangency. These right angles are key to understanding the relationship between the angle formed by the tangents and the angle subtended at the center of the circle.

Triangle Formation and External Angle Theorem
The two radii and the line segment joining the points of tangency form a triangle within the circle. The angle between the two tangents, drawn from the external point, acts as an external angle to this triangle. According to the external angle theorem in geometry, an external angle of a triangle is equal to the sum of the two opposite internal angles.

In our triangle, the two internal angles are right angles (90° each) because they are formed by the radii perpendicular to the tangents. The sum of these right angles is 180°. Therefore, the external angle, which is the angle between the two tangents, is also 180°. This means that the angle between the tangents is supplementary to the angle subtended by the line segment at the center of the circle.

Thus, we conclude that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center. This theorem elegantly illustrates the harmonious relationships in circle geometry, highlighting how tangents, radii, and angles cohesively interact to form consistent and predictable geometric patterns. It is a testament to the beauty and logical structure of mathematics in the realm of geometry.

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Questions of 10th Maths Exercise 10.2 in Detail