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[[File:Cyclic quadrilateral.jpg|alt=Fig. 1|thumb|Fig. 1]] | |||
A quadrilateral ABCD is called cyclic if all the four vertices | |||
of it lie on a circle (see Fig 1). | |||
Here we find that <math>\angle A+\angle C =180^\circ</math>and <math>\angle B+\angle D =180^\circ</math>. | |||
Theorems related to Cyclic Quadrilateral are mentioned below. | |||
Theorem 1: The sum of either pair of opposite angles of a cyclic quadrilateral | |||
is <math>180^\circ</math>. | |||
Theorem 2: If the sum of a pair of opposite angles of a quadrilateral is <math>180^\circ</math>, | |||
the quadrilateral is cyclic. | |||
== Examples == | |||
[[File:Cyclic quadrilateral - 2.jpg|alt=Fig. 2|thumb|Fig. 2]] | |||
1: In Fig 2, <math>ABCD</math> is a cyclic quadrilateral in which <math>AC</math> and <math>BD</math> are its diagonals. | |||
If <math>\angle DBC =55^\circ</math>and <math>\angle BAC =45^\circ</math>, find <math>\angle BCD</math> | |||
Solution: | |||
<math>\angle CAD=\angle DBC =55^\circ</math>(Angles in the same segment) | |||
Therefore, <math>\angle DAB=\angle CAD+\angle BAC</math> | |||
<math>\angle DAB=55^\circ +45^\circ=100^\circ</math> | |||
But <math>\angle DAB+\angle BCD = 180^\circ</math>(Opposite angles of a cyclic quadrilateral) | |||
<math>\angle BCD = 180^\circ -\angle DAB</math> | |||
<math>\angle BCD = 180^\circ -100^\circ =80^\circ</math> | |||
[[Category:वृत्त]][[Category:कक्षा-9]][[Category:गणित]] | [[Category:वृत्त]][[Category:कक्षा-9]][[Category:गणित]] | ||
Revision as of 07:15, 11 September 2024
A quadrilateral ABCD is called cyclic if all the four vertices
of it lie on a circle (see Fig 1).
Here we find that and .
Theorems related to Cyclic Quadrilateral are mentioned below.
Theorem 1: The sum of either pair of opposite angles of a cyclic quadrilateral
is .
Theorem 2: If the sum of a pair of opposite angles of a quadrilateral is ,
the quadrilateral is cyclic.
Examples
1: In Fig 2, is a cyclic quadrilateral in which and are its diagonals.
If and , find
Solution:
(Angles in the same segment)
Therefore,
But (Opposite angles of a cyclic quadrilateral)