In a circle a diameter $A B$ $AB$ is drawn and on one side of it an arc $C D$ $CD$ is marked, which subtends an angle $50^{\circ}$ $50^@$ at the center. $A C$ $AC$ and $B D$ $BD$ are joined and produced to meet at $E$ $E$ . Find $∠ C E D$ $/_CED$ ?

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In a circle a diameter $A B$ $AB$ is drawn and on one side of it an arc $C D$ $CD$ is marked, which subtends an angle $50^{\circ}$ $50^@$ at the center. $A C$ $AC$ and $B D$ $BD$ are joined and produced to meet at $E$ $E$ . Find $∠ C E D$ $/_CED$ ?

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Answer 1 · 2017-02-02T08:45:07

$∠ C E D = 65^{\circ}$ $/_CED=65^@$

Explanation:

Consider the following figure, as described in the question, where we have joined $A D$ $AD$ and $B C$ $BC$ and let $∠ B O D = x$ $/_BOD=x$ and $∠ A O C = y$ $/_AOC=y$ .
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As chord $B C$ $BC$ subtends angle $∠ B O C$ $/_BOC$ at center and $∠ B A E$ $/_BAE$ at the circle, we have $∠ B A E = \frac{1}{2} (50 + x) = 25 + \frac{x}{2}$ $/_BAE=1/2(50+x)=25+x/2$

Similarly for chord $A D$ $AD$ , we have $∠ A B E = \frac{1}{2} (50 + y) = 25 + \frac{y}{2}$ $/_ABE=1/2(50+y)=25+y/2$

Hence $∠ C E D$ $/_CED$ , being the third angle of $Δ A B E$ $DeltaABE$ is given by

$∠ C E D = 180^{\circ} - ∠ B A E - ∠ A B E$ $/_CED=180^@-/_BAE-/_ABE$

= $180^{\circ} - 25^{\circ} - \frac{x}{2} - 25^{\circ} - \frac{y}{2} = 130^{\circ} - \frac{x + y}{2}$ $180^@-25^@-x/2-25^@-y/2=130^@-(x+y)/2$

= $130^{\circ} - \frac{1}{2} (180^{\circ} - 50^{\circ}) = 130^{\circ} - 65^{\circ} = 65^{\circ}$ $130^@-1/2(180^@-50^@)=130^@-65^@=65^@$

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In a circle a diameter $A B$ $AB$ is drawn and on one side of it an arc $C D$ $CD$ is marked, which subtends an angle $50^{\circ}$ $50^@$ at the center. $A C$ $AC$ and $B D$ $BD$ are joined and produced to meet at $E$ $E$ . Find $∠ C E D$ $/_CED$ ?

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Explanation:

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In a circle a diameter ABABAB is drawn and on one side of it an arc CDCDCD is marked, which subtends an angle 50^@50∘50^@ at the center. ACACAC and BDBDBD are joined and produced to meet at EEE. Find /_CED∠CED/_CED?

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In a circle a diameter $A B$ $AB$ is drawn and on one side of it an arc $C D$ $CD$ is marked, which subtends an angle $50^{\circ}$ $50^@$ at the center. $A C$ $AC$ and $B D$ $BD$ are joined and produced to meet at $E$ $E$ . Find $∠ C E D$ $/_CED$ ?