Question

In the figure given below, it is given that AE = AD and BD = CE. Prove that ∆AEB ≌ ∆ADC.

Answer 1

Hint: Before solving this question, we must know the different ways to prove congruence:-
SIDE – SIDE – SIDE (SSS): If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule.
SIDE – ANGLE – SIDE (SAS): If any two sides and angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.
ANGLE – SIDE – ANGLE (ASA): If any two angles and sides included between the angles of one triangle are equivalent to the corresponding two angles and sides included between the angles of the second triangle, then the two triangles are said to be congruent by ASA rule.
RIGHT ANGLE – HYPOTENUSE – SIDE (RHS): If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent by RHS rule.

Complete step-by-step solution -
Given: AE = AD
BD = CE
To prove: ∆AEB ≌ ∆ADC
Proof:-

Let us firstly prove the congruence of ∆EBC and ∆DCB.
CB = CB (common)
Angle ECB = Angle DBC (base angles of an isosceles triangle are equal)
EC = DB (given)
Therefore, ∆EBC and ∆DCB are congruent by SAS congruence.
Now, angle DCB = angle EBC (through C.P.C.T.)
As we know that angle ACB and angle ABC are equal angles as they are the base angles of an isosceles triangle, so, we can write them as:-
Angle ACB = angle ABC
Angle ACD + Angle DCB = Angle ABE + Angle EBC
Angle ACD = Angle ABE (because angle DCB = angle EBC (through C.P.C.T.))

Now, we will prove the congruence of ∆AEB and ∆ACD.
AC = AB (AE + EC = AD + DB) (given)
Angle A = Angle A (common angle)
Angle ACD = Angle ABE (proved above)
Hence, ∆AEB ≌ ∆ADC by ASA congruence


Note:- Let us now know about C.P.C.T.
C.P.C.T.: CPCT stands for Corresponding parts of congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.