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7. $\small AB$ is a line segment and $\small P$ is its mid-point. $\small D$ and $\small E$ are points on the same side of $\small AB$ such that $\small \angle BAD=\angle ABE$ and $\small \angle EPA=\angle DPB$ (see Fig). Show that

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From the figure, it is clear that : $\angle EPA\ =\ \angle DPB$

Adding $\angle DPE$ both sides, we get : $\angle EPA\ +\ \angle DPE =\ \angle DPB\ +\ \angle DPE$

$\angle DPA =\ \angle EPB$

Now, consider $\Delta DAP$ and $\Delta EBP$ :

(i) $\angle DPA =\ \angle EPB$ (Proved above)

(ii) $AP\ =\ BP$ (Since P is the midpoint of line AB)

(iii) $\small \angle BAD=\angle ABE$ (Given)

Hence by ASA congruence, we can say that : $\small \Delta DAP\cong \Delta EBP$

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Devendra Khairwa

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