Question

In the given figure AB = BC, AD = CD Prove that ZADE is a right angle and AE and EC are equal. ​

Answer 1

In triangle ABD and CBD,

AB = BC (Given)

AD = CD (Given)

BD = BD (Common side)

Hence, Triangle ABD and CBD are congruent by SSS criteria.

∠ADB = ∠CDB (CPCT)

Also,

∠ADB + ∠CDB = 180° (Linear Pair)

=> ∠ADB + ∠ADB = 180°

=> 2∠ADB = 180°  (∠ADB = ∠CDB)

=> ∠ADB = 90°

∠ADB + ∠ADE = 180° (Linear pair)

=> 90° + ∠ADE = 180°

=> ∠ADE = 90°

Hence, ∠ADE is a right angle

Similarly, as ∠ADE and ∠EDC form a linear pair, ∠EDC is 90°

In triangles ADE and EDC,

AD = DC (Given)

∠ADE = ∠EDC = 90° (Proved above)

ED = ED (Common side)

Hence, triangles ADE and EDC are congruent by SAS criteria

Therefore, AE = EC (CPCT)

PLEASE MARK AS BRAINLIEST

Answer 2

Answer:

given: in the triangle ABC ; AB = BC and AD = DC

TPT : ∠ADE=90 and AE=EC

proof:

in the triangle ADB and CDB;

AB = BC (given)

AD = CD (given)

and BD is common.

therefore by SSS congruency ; △ADB≅△CDB

therefore by CPCT congruency property;

∠ABD=∠CBD ............(1)and ∠ADB=∠CDB=1802=90

∠ADE=∠CDB [vertically opposite angles]∠ADE=90 deg

now in the triangles ABE and CBE;

AB=BC [given]∠ABE=∠CBE [from eq(1)]BE is common.therefore by SAS congruency△ABE≅△CBE

and by cpct, AE = EC.

HENCE PROVED