Question

In the figure AD = BC and BD = CA .Prove that ∠ADB = ∠BCA and ∠DAB = ∠CBA.

Answer 1

Hint: Start by noting the given information and draw the diagram. Take two appropriate triangles which would help in leading to the desired answer . Check for the congruency between these two triangles and then apply CPCT in order to get the desired result.

Complete step-by-step answer:
Given,
AD = BC
BD = CA
To Prove- ∠ADB = ∠BCA and ∠DAB = ∠CBA.
Proof- Let us take ∆ADB and ∆BCA into consideration .

AD = BC (Given)
BD = AC (Given)
AB = AB (Common side)
∴∆ADB ≅ ∆BCA by SSS
SSS(side-side-side) congruence states that if two triangles have all the three sides equal they are congruent.
Now , Applying CPCT( Corresponding parts of congruent triangles) which states that, If two triangles are congruent then their corresponding sides are equal , also corresponding angles too.
∠ADB = ∠BCA
Similarly , ∠DAB = ∠CBA
Hence proved.

Note: All the postulates of congruence must be known in order to solve such similar problems involving congruent triangles . Attention must be given while selecting the triangles to be proved congruent. Also property such as CPCT must be well known.