Question

ABCD is a rhombus in which the altitude from D to side AB bisects AB. Then AngleA and AngleB respectively, are _______ .

Answer 1

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In a ΔAED and ΔBED,
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DE = DE ( common line)

∠AED = ∠BED ( right angle)
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AE = EB ( DE is an altitude)

∴ ΔAED ≅ ΔBED ( SAS property)
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∴ AD = BD ( by C.P.C.T)

But AD = AB ( sides of rhombus are equal)

⇒ AD = AB = BD
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∴ ABD is an equilateral traingle.

∴ ∠A = 60°

⇒ ∠A = ∠C = 60° (opposite angles of rhombus are equal)
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But Sum of adjacent angles of a rhombus is supplimentary.

∠ABC + ∠BCD = 180°

⇒ ∠ABC + 60°= 180°
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⇒ ∠ABC = 180° - 60° = 120°.

∴ ∠ABC = ∠ADC = 120°.(opposite angles of rhombus are equal)
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∴ Angles of rhombus are ∠A = 60° and ∠C = 60° , ∠B = ∠D = 120°.