![]() ![]() Circle theorems are statements in geometry that state important results related to circles that are used to solve various questions in geometry.So, we have AB = CD (Corresponding parts of congruent triangles). So, triangles AOB and COD are congruent by SAS congruence rule. Proof: Consider a circle given below with center O and two chords AB and CD such that ∠AOB = ∠COD. Theorem 5: If the angles subtended by two chords at the center are equal, then the two chords are equal. So, we have ∠AOB = ∠COD (Corresponding parts of congruent triangles). So, triangles AOB and COD are congruent by SSS congruence rule. Proof: Consider a circle given below with center O and two chords AB and CD such that AB = CD. Theorem 4: Two equal chords subtend equal angles at the center of the circle. Hence we have proved the circle theorem 'The angles subtended at the circumference by the same arc are equal.' Since angles ACB and ADB are arbitrary angles, therefore, the result is true for all angles subtended by the same arc. Using the circle theorem 'The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.', we have thatįrom equations (1) and (2), we get ∠ACB= ∠ADB. Proof: Consider the following figure, which shows an arc AB subtending angles ACB and ADB at two arbitrary points C and D on the circumference. Theorem 3: The angles subtended at the circumference by the same arc are equal. Hence we have proved the circle theorem 'The angle subtended by the diameter at the circumference is a right angle'. Now, ∠AOB = 180° as AB is a straight line (diameter). Using theorem 1 'The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.', we have ∠AOB= 2× ∠ACB. Proof: Consider the figure below, where AB is the diameter of the circle. Theorem 2: The angle subtended by the diameter at the circumference is a right angle. Hence, we have proved the circle theorem 'The angle subtended by a chord at the center is twice the angle subtended by it at the circumference.' Hence, using the exterior angle theorem, we get, Angles opposite to equal sides are equal) In ΔOBC, ∠OBC = ∠OCB, because OB = OC (OB and OC being the radii.In ΔOAC, ∠OAC = ∠OCA, because OA = OC (OA and OC being the radii.There are two triangles formed ΔOAC and ΔOBC. Draw a line segment through O and C, and let it intersect the circle again at point D, as shown. Proof: Consider the following circle, in which an arc (or segment) AB subtends ∠AOB at the center O and ∠ACB at a point C on the circumference. Theorem 1: The angle subtended by a chord at the center is twice the angle subtended by it at the circumference. In this section, let us prove some of the important circle theorems discussed above. Let us now go through the statements of some important circle theorems in the next section. Arc: An arc of a circle is referred to as a curve, which is a part or portion of its circumference/boundary.Sector: A sector of a circle is the area enclosed by the two radii and arc of the circle.Segment: A segment of a circle is the area enclosed by a chord and arc of a circle.Tangent: A tangent to a circle is a line segment that touches the circle at a unique point and lies outside the circle.Circumference: The circumference of a circle is the perimeter of its boundary. ![]()
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