ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that
(i) ΔABD ≅ ΔBAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.
Answers
Step-by-step explanation:
(i)In quadrilateral ABCD
angle DAB=angleCBA (given)
AD=BC (given)
AB=BA (common)
(ii)BD =AC (cpct)
(iii)angleABD=angleBAC (cpct)
Answer:
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Congruence of triangles:
Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.
In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.
It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.
Criteria for congruence of triangles:
There are 4 criteria for congruence of triangles.
SAS( side angle side):
Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.
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First use, SAS rule to show congruence of triangles and then use CPCT to show ii & iii parts.
Given:
In quadrilateral ABCD,
AD = BC &
∠DAB = ∠CBA
To Prove:
(i) ΔABD ≅ ΔBAC
(ii) BD=AC
(iii) ∠ABD = ∠BAC
Proof:
i)
In ΔABD & ΔBAC,
AB = BA (Common)
∠DAB = ∠CBA (Given)
AD = BC (Given)
Hence, ΔABD ≅ ΔBAC.
( by SAS congruence rule).
(ii) Since, ΔABD ≅ ΔBAC
Then, BD = AC ( by CPCT)
(iv) Since, ΔABD ≅ ΔBAC
Then , ∠ABD = ∠BAC (by CPCT)