The opposite
angles of a cyclic quadrilateral are supplementary.
Given: □ ABCD is a cyclic quadrilateral.
To Prove: ∠ A + ∠ C = 1800
and ∠ B + ∠ D = 1800
Solution:
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∠ A = ½ m(arc BCD) --------------eq. no. (1)
[Inscribed
angle theorem]
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∠ C = ½ m (arc BAD) -------------- eq. no. (2)
[Inscribed
angle theorem]
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Adding equations (1) and (2)
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∴
∠
A + ∠
C = ½
m(arc BCD) + ½ m(arc BAD)
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∴
∠
A + ∠
C = ½
[m(arc BCD) + m(arc BAD)]
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∴
∠
A + ∠
C = ½
[Measure of a circle]
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∴
∠
A + ∠
C = ½
[3600]
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∴
∠
A + ∠
C = 1800
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Similarly, we can prove that,
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∠
B + ∠
D = 1800
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