Question

ABCD is a square and ABE is an equilateral triangle outside the square prove that the angle ACE is equal to half angle ABC​

Answer 1

Answer:

Step-by-step explanation:

As ABE is an equilateral triangle, hence  

∠ABE=600 .......(i)

AB=BE

And ABCD is a square hence

∠ABC=900

AB=BC........(ii)

By (i) and (ii) we get

BC=BE

In triangle BCE

∠BEC=∠BCE [angles opposite to equal sides are equal]

Also

⟹∠BEC+∠BCE+∠CBE=1800 [Angle sum property]

⟹∠BCE+∠BCE+(∠ABC+∠ABE)=1800

2⟹∠BCE+(900+600)=1800

2⟹∠BCE=1800−1500

⟹∠BCE=150

As diagonals of the square bisect the interior angle, hence

∠ACD=9002=450

∠BCD=∠ACD+∠ACE+∠BCE

⟹900=450+∠ACE+150

⟹∠ACE=900−600

⟹∠ACE=300

⟹∠ACE=12(600)

⟹∠ACE=12(∠ABE)