In the figure given below two chords $P Q$ and $X Y$ of the same circle are parallel to each other. $O$ is the centre of the circle. Show that $∠ P O X ≅ ∠ Q O Y$ .

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Solution

Draw seg $P Y$ ,
seg $P Q ∥$ seg $X Y$ and seg $P Y$ is their transversal.
$∴ ∠ X Y P = ∠ Y P Q$ ....(alternate angles) ...(1)
Inscribed angle, $∠ Y P Q$ intercepts arc $Y Q$ .
$∴ ∠ Y P Q = \frac{1}{2} m (a r c Y Q)$ ....(by inscribed angle theorem) ...(2)
Similarly, $∠ X Y P = \frac{1}{2} m (a r c X P)$ ....(3)
$∴ \frac{1}{2} m (a r c X P) = \frac{1}{2} m (a r c Y Q)$ ....[from (1), (2) and (3)]
$∴ m (a r c X P) = m (a r c Y Q)$
Now, $m (a r c X P) = ∠ X O P$
$m (a r c Y Q) = ∠ Y O Q$
$∴ ∠ X O P = ∠ Y O Q$ ....(4)
$∴ ∠ X O P = ∠ Y O Q$
i.e. $∠ P O X = ∠ Q O Y$

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In the figure given below two chords PQ and XY of the same circle are parallel to each other. O is the centre of the circle. Show that ∠POX≅∠QOY.

In the figure given below two chords $P Q$ and $X Y$ of the same circle are parallel to each other. $O$ is the centre of the circle. Show that $∠ P O X ≅ ∠ Q O Y$ .