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ABCD is a square and ΔBCE is an equilateral triangle such that E is outside the square. Find ∠AED.


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Answer 1

Answer:

∠AED = 30°

Step-by-step explanation:

Draw the figure :

• ABCD is a square with all sides equal .

• On side BC , draw an equilateral Δ .

• Every angle of Δ BCE is 60°

In Δ ABE and Δ CED we note the following :

AB = CD [ side of square ]

BE = CE [ side of equilateral Δ ]

∠ABE = 90° + 60°

[ 90° because every ∠ of square is 90° and 60° because every ∠ of equilateral Δ is 60° ]

∠ABE = 150°

∠ECD = 90° + 60° [ Same reason ]

∠ECD = 150°

Hence ∠ABE = ∠ECD

Thus Δ ABE ≅ Δ ECD [ S.A.S criteria ]

We can now say that AE = ED [ c.p.c.t ]

In Δ ABE ,

AB = EB

[ EB = BC ( equilateral Δ ) and AB = BC ( side of square ) ]

Hence ∠BAE = ∠BEA [ IsoscelesΔ ]

∠BAE + ∠BEA + ∠ABE = 180°

= > 2 ∠BAE + 150° = 180°

= > 2 ∠BAE = 180° - 150°

= > 2 ∠BAE = 30°

= > ∠BAE = 15°

We know that ∠BAD = 90° [ ∠ of a square ]

= > ∠EAD = 90° - 15°

= > ∠EAD = 75°

We had already proved AE = ED

So ∠ EAD = ∠ EDA [ Isosceles Δ ]

= > ∠ EDA = 75°

Now by ∠ sum property we have :

∠EAD + ∠AED + ∠ADE = 180°

= > ∠AED + 75° + 75° = 180°

= > ∠AED + 150° = 180°

= > ∠AED = 180° - 150°

= > ∠AED = 30°

Answer 2

On side BC , draw an equilateral Δ .

Every angle of Δ BCE is 60°

In Δ ABE and Δ CED we find

AB = CD (side of square )

BE = CE ( side of equilateral Δ )

∠ABE = 90° + 60°
(90° because every ∠ of square is 90° and 60° because every ∠ of equilateral Δ is 60° )

∠ABE = 150°

∠ECD = 90° + 60° ( Same reason )

∠ECD = 150°

Hence ∠ABE = ∠ECD

so, Δ ABE ≅ Δ ECD (S.A.S criteria )

now ,
AE = ED [ c.p.c.t ]

In Δ ABE ,

AB = EB
(EB = BC ( equilateral Δ ) and AB = BC ( side of square ) )

Hence ∠BAE = ∠BEA
(IsoscelesΔ )

∠BAE + ∠BEA + ∠ABE = 180°
=> 2 ∠BAE + 150° = 180°
=> 2 ∠BAE = 180° - 150°
=> 2 ∠BAE = 30°
=> ∠BAE = 15°

We know that ∠BAD = 90° (∠ of a square)

=> ∠EAD = 90° - 15°
=> ∠EAD = 75°

We had already proved AE = ED
So ∠ EAD = ∠ EDA [ Isosceles Δ ]
=> ∠ EDA = 75°

Now by ∠ sum property we have :
∠EAD + ∠AED + ∠ADE = 180°
=> ∠AED + 75° + 75° = 180°
=> ∠AED + 150° = 180°
=> ∠AED = 180° - 150°
=> ∠AED = 30°

hope u understand ❤️❤️❤️❤️