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# In △ABC and △DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. 8.22). Show that

(i) quadrilateral ABED is a parallelogram

(ii) quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) quadrilateral ACFD is a parallelogram

(v) AC = DF

(vi) △ABC ≅ △DEF.

**Solution:**

Given: In ΔABC and ΔDEF, AB = DE, AB || DE, BC = EF and BC || EF.

We can use the fact that in a quadrilateral if one pair of opposite sides are parallel and equal to each other then it will be a parallelogram.

**(i)** It is given that AB = DE and AB || DE

If one pair of opposite sides of a quadrilateral are equal and parallel to each other, then it will be a parallelogram.

Therefore, quadrilateral ABED is a parallelogram.

**(ii)** It is given that BC = EF and BC || EF

Therefore, quadrilateral BEFC is a parallelogram.

**(iii)** As we had observed that ABED and BEFC are parallelograms, therefore

AD = BE and AD || BE (Opposite sides of a parallelogram are equal and parallel)

BE = CF and BE || CF (Opposite sides of a parallelogram are equal and parallel)

Thus, AD = BE = CF and AD || BE || CF

∴ AD = CF and AD || CF (Lines parallel to the same line are parallel to each other)

**(iv)** As we had observed that one pair of opposite sides (AD and CF) of quadrilateral ACFD are equal and parallel to each other, therefore, it is a parallelogram.

**(v)** As ACFD is a parallelogram, therefore, the pair of opposite sides will be equal and parallel to each other

∴ AC || DF and AC = DF

**(vi)** ∆ABC and ∆DEF,

AB = DE (Given)

BC = EF (Given)

AC = DF (Since ACFD is a parallelogram)

∴ ∆ABC ≅ ∆DEF (By SSS congruence rule)

**☛ Check: **Class 9 Maths NCERT Solutions Chapter 8

**Video Solution:**

## In △ABC and △DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. 8.22). Show that (i) quadrilateral ABED is a parallelogram (ii) quadrilateral BEFC is a parallelogram (iii) AD || CF and AD = CF (iv) quadrilateral ACFD is a parallelogram (v) AC = DF (vi) △ABC ≅ △DEF.

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 11

**Summary:**

If in ΔABC and ΔDEF, AB || DE, BC = EF and BC || EF, vertices A, B, C are joined to vertices D, E, F respectively, then quadrilateral ABED is a parallelogram, quadrilateral BEFC is a parallelogram, AD || CF and AD = CF, quadrilateral ACFD is a parallelogram, AC = DF, and △ABC ≅ △DEF.

**☛ Related Questions:**

- ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
- ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:(i) ABCD is a square(ii) diagonal BD bisects ∠B as well as ∠D.
- In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram
- ABCD is a trapezium in which AB || CD and AD = BC (see the given figure). Show that i) ∠A = ∠B ii) ∠C = ∠D iii) ∆ABC ≅ ∆BAD iv) diagonal AC = diagonal BD [Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

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