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.]