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X and Y are points on the side LN of the...

X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (see figure). Prove that `ar (DeltaLZY) = ar (MZYX)`.

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We have to prove that `ar(Delta LZY) = ar(MZYX)`
Since, `Delta LXZ and Delta XMZ` are on the same base the between the same parallels LM and XZ, we have
`ar(DeltaLXZ) = ar(DeltaXMZ)`.....(1)
Adding `ar(Delta XYZ)` to both sides of (1), we get
`ar(Delta LXZ) + ar(Delta XYZ) = ar (Delta XMZ) + ar(Delta XYZ)`
`rArr ar(DeltaLZY) = ar(MZYX)`
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