Mathematics
AB is a line segment. AX and BY are two equal line segments drawn on opposite sides of AB such that AX || YB. If AB and XY intersect at M, prove that :
(i) △AMX ≅ △BMY
(ii) AB and XY bisect each other at M.
Triangles
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Answer
(i) In △AMX and △BMY,
⇒ ∠AXM = ∠BYM [Alternate angles are equal as AX || YB]
⇒ AX = BY [Given]
⇒ ∠AMX = ∠BMY [Vertically opposite angles are equal]
∴ △AMX ≅ △BMY (By A.A.S axiom)
Hence, proved that △AMX ≅ △BMY.
(ii) As, △AMX ≅ △BMY
⇒ AM = MB [Corresponding parts of congruent triangles are equal]
∴ M is the mid-point of line segment AB.
⇒ XM = MY [Corresponding parts of congruent triangles are equal]
∴ M is the mid-point of line segment XY.
Hence, proved that AB and XY bisect each other at M.
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