In cyclic quadrilateral $A B C D$ , if $∠ A : ∠ C = 5 : 4$ , find $∠ A$ .

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Solution

Given, cyclic quadrilaterals $A B C D$ .
and $∠ A : ∠ C = 5 : 4$ .
Let the $∠ A = 5 x$ , $∠ B = 4 x$ .
We know, in a cyclic quadrilateral, the sum of opposite angles are supplementary, i.e. $180^{o}$ .
Then, $∠ A + ∠ C = 180^{o}$
$⟹$ $5 x + 4 x = 180^{o}$
$⟹$ $9 x = 180^{o}$
$⟹$ $x = 20^{o}$ .
Now,
$∠ A = 5 x$ $= 5 \times 20^{o}$ $= 100^{o}$ .
Hence, $∠ A = 100^{o}$ .

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In cyclic quadrilateral ABCD, if ∠A:∠C=5:4, find ∠A.

Given, cyclic quadrilaterals ABCD.and ∠A:∠C=5:4.Let the ∠A=5x, ∠B=4x.We know, in a cyclic quadrilateral, the sum of opposite angles are supplementary, i.e. 180o.Then, ∠A+∠C=180o⟹ 5x+4x=180o⟹ 9x=180o⟹ x=20o.Now,∠A=5x =5×20o =100o.Hence, ∠A=100o.

In cyclic quadrilateral $A B C D$ , if $∠ A : ∠ C = 5 : 4$ , find $∠ A$ .