Question

Prove the Question according to the theorem of a Circle

Answer 1

Given -

P,Q,R and S are 4 points on the circle and PQRS is a cyclic quadrilateral

Prove -

`\angle PQR\text{ + }\angle PSR\text{ = 180}`

Explanation -

`\angle1\text{ = }\angle6\text{ ------\lparen1\rparen \lparen Angles in same segment\rparen}`
`\angle5\text{ = }\angle8\text{ ------\lparen2\rparen \lparen Angles in the same segment\rparen}`
`\angle2\text{ = }\angle8\text{ ------\lparen3\rparen \lparen Angles in the same segment\rparen}`
`\angle7\text{ = }\angle3\text{ -------\lparen4\rparen\lparen Angles in the same segment\rparen}`

By using angle sum property of quadrilateral

`\angle P\text{ + }\angle Q\text{ + }\angle R\text{ + }\angle S\text{ = 360}`
`\angle1\text{ + }\angle2\text{ + }\angle3\text{ + }\angle4\text{ + }\angle5\text{ + }\angle6\text{ + }\angle7\text{ + }\angle8\text{ = 360}`
`(\angle1+\angle2+\angle7+\angle8)+(\angle3+\angle4+\angle5+\angle6)=360`

By using equation 1,2,3 and 4

`2(\angle3+\angle4+\angle5+\angle6)\text{ = 360}`
`\angle3+\angle4+\angle5+\angle6\text{ = 180}`
`(\angle3+\angle4)+(\angle5+\angle6)\text{ = 180}`
`\angle PQR\text{ + }\angle PSR\text{ = 180}`

Hence Proved