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.
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.
Related Questions
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In the given figure, PA ⊥ AB; QB ⊥ AB and PA = QB. If PQ intersects AB at M, show that M is the mid-point of both AB and PQ.
In the given figure, the sides BA and CA of △ABC have been produced to D and E such that BA = AD and CA = AE. Prove that, ED || BC.
In the given figure, the line segments AB and CD intersect at a point M in such a way that AM = MD and CM = MB. Prove that, AC = BD but AC may not be parallel to BD.
