Prove the Following:

Question

Given isosceles triangle ABC with base AB, angle CAD is congruent to angle CBE, and triangle ABE is congruent to triangle BAD. Prove that triangle ACD is congruent to triangle BCE

Answer 1

Since ABC is an isoceles triangle with base AB,

AC = BC by defintion

angles CAB = CBA are congruent base angles of

this isocoeles triangle

By angle addition postulate,

angle CAB = angle CAD+angle BAD

AND

angle CBA = angle CBE + angle EBA

Then by substitution/transitive:

angle CAD+angle BAD = angle CBE + angle EBA

However, angle CAD = angle CBE is given

so angle BAD = angle EBA by subtraction property of equality

In summary:

AC = BC

angle CAD = angle CBE

and angle C = angle ACD = angle BCE by reflexive property

the result of the proof follows by ASA