# ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that

i) △ ABD ≅ △ BAC (ii) BD = AC (iii) ∠ABD = ∠BAC.

**Solution:**

Given: AD = BC and ∠DAB = ∠CBA

To Prove: i) △ ABD ≅ △ BAC ii) BD = AC iii) ∠ABD = ∠BAC

(i) In △ABD and △BAC,

AD = BC (Given)

∠DAB = ∠CBA (Given)

AB = BA (Common)

∴ △ABD ≅ △BAC (By SAS congruence rule)

(ii) Since △ABD ≅ △BAC,

∴ BD = AC (By CPCT)

(iii) Since △ABD ≅ △BAC,

∠ABD = ∠BAC (By CPCT)

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 7

**Video Solution:**

## ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that i) △ ABD ≅ △ BAC ii) BD = AC iii) ∠ABD = ∠BAC

NCERT Maths Solutions Class 9 Chapter 7 Exercise 7.1 Question 2

**Summary:**

If ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA, then ΔABD ≅ ΔBAC using SAS congruence criteria which implies BD = AC, and ∠ABD = ∠BAC.

**☛ Related Questions:**

- In quadrilateral ACBD, AC = AD and AB bisects ∠A (See the given figure).Show that Δ ABC ≅ Δ ABD. What can you say about BC and BD?
- AD and BC are equal perpendiculars to a line segment AB (See the given figure). Show that CD bisects AB.
- l and m are two parallel lines intersected by another pair of parallel lines p and q. Show that ΔABC ≅ ΔCDA.
- Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see the given figure). Show that:i) ΔAPB ≅ ΔAQBii) BP = BQ or B is equidistant from the arms of ∠A

Math worksheets and

visual curriculum

visual curriculum