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Equilateral triangle example

Question: ABC is an equilateral triangle. AD is perpendicular to BC at point D. Extended AB to the point E to make EB = DB. Prove AD = ED.

Proof:
Because ABC is an equilateral triangle (given),
so, angle ABC = angle C = angle CAB = 60o
(Each interior angle in an equilateral triangle is 60 degrees.)
Because AD perpendicular to BC at point D (given),
so, DB = CD and
angle BAD = angle CAD = (1/2) angle CAB = (1/2) 60o = 30o
(The altitude of an isosceles triangle bisects its vertex and its base.)
so, BD = DC = (1/2)BC
Because angle ABD is an external angle of the triangle BED,
so, angle ABD = angle E + angle BDE.
(external angle of a triangle theorem)
Because BE = BD (given),
so, angle E = angle BDE
(In a triangle, congruent sides opposite congruent angles.)
angle E = angle BDE = (1/2) angle ABD = (1/2) 60o = 30o
In triangle AED, because angle E = angle DAE, so ED = AD
(In a triangle, congruent angles opposite congruent sides.)