ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If $∠ A D E = 70^{\circ}$ and $∠ O B A = 45^{\circ}$ , calculate $∠ B A C$ and $∠ A O C$

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Solution

ABCD is a cyclic quadrilateral with center O.
Therefore OA=OA=OC are the radii of circle
In $△ A O B$ , OA=OB, therefore
$∠ A B O = ∠ B A O = ∠ B A C = 45^{\circ}$ [Angles opposite to equal sides]
$∠ O C A -$ cannot be determined. Because
$A O C$ is a line, $∠ A O C = 180^{\circ}$

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ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If ∠ADE=70∘ and ∠OBA=45∘, calculate ∠BAC and ∠AOC

ABCD is a cyclic quadrilateral with center O.Therefore OA=OA=OC are the radii of circleIn △AOB, OA=OB, therefore∠ABO=∠BAO=∠BAC=45∘ [Angles opposite to equal sides]∠OCA− cannot be determined. BecauseAOC is a line, ∠AOC=180∘

ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If $∠ A D E = 70^{\circ}$ and $∠ O B A = 45^{\circ}$ , calculate $∠ B A C$ and $∠ A O C$

ABCD is a cyclic quadrilateral with center O.
Therefore OA=OA=OC are the radii of circle
In $△ A O B$ , OA=OB, therefore
$∠ A B O = ∠ B A O = ∠ B A C = 45^{\circ}$ [Angles opposite to equal sides]
$∠ O C A -$ cannot be determined. Because
$A O C$ is a line, $∠ A O C = 180^{\circ}$