ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (see figure). Prove that
NCERT Solutions for Class 9 Maths Chapter 7 Triangles Ex 7.1 Q2
(i) ∆ABD ≅ ∆BAC
(ii) BD = AC
(iii) ∠ABD = ∠ BAC
Answers
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.
----------------------------------------------------------------------------------------------------
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)
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.
----------------------------------------------------------------------------------------------------
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)
===========================================
Hope this will help you..
Answer:
In △ABD and △BAC,
AD=BC (Given)
∠DAB=∠CBA (Given)
AB=BA (Common)
∴△ABD≅△BAC (By SAS congruence rule)
∴BD=AC (By CPCT)
And, ∠ABD=∠BAC (By CPCT)