Chapter 8 Quadrilaterals Ex 8.1 Solutions
Question - 11 : - 11. In ╬ФABC and╬ФDEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joinedto vertices D, E and F respectively (see Fig. 8.22).
Show that
(i) quadrilateralABED is a parallelogram
(ii) quadrilateralBEFC is a parallelogram
(iii) AD || CF andAD = CF
(iv) quadrilateralACFD is a parallelogram
(v) AC = DF
(vi) ╬ФABC тЙЕ ╬ФDEF.
Answer - 11 : -
(i) AB = DE and AB || DE (Given)
Two opposite sides of a quadrilateral are equal and parallel toeach other.
Thus, quadrilateral ABED is a parallelogram
(ii) Again BC = EF and BC || EF.
Thus, quadrilateral BEFC is a parallelogram.
(iii) Since ABED and BEFC are parallelograms.
тЗТ AD =BE and BE = CF (Opposite sides of a parallelogram are equal)
, AD = CF.
Also, AD || BE and BE || CF (Opposite sides of a parallelogramare parallel)
, AD || CF
(iv) AD and CF are opposite sides of quadrilateral ACFD whichare equal and parallel to each other. Thus, it is a parallelogram.
(v) Since ACFD is a parallelogram
AC || DF and AC = DF
(vi) In ╬ФABC and ╬ФDEF,
AB = DE (Given)
BC = EF (Given)
AC = DF (Opposite sides of a parallelogram)
, ╬ФABC тЙЕ ╬ФDEF[SSS congruency]
Question - 12 : - 12. ABCD is atrapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that
(i) тИаA = тИаB
(ii) тИаC = тИаD
(iii) ╬ФABC тЙЕ ╬ФBAD
(iv) diagonal AC =diagonal BD
[Hint : Extend ABand draw a line through C parallel to DA intersecting AB produced at E.]
Answer - 12 : -
To Construct: Draw a line through C parallel to DA intersectingAB produced at E.
(i) CE = AD (Opposite sides of a parallelogram)
AD = BC (Given)
, BC = CE
тЗТтИаCBE = тИаCEB
also,
тИаA+тИаCBE = 180┬░ (Angles on the same side oftransversal and тИаCBE = тИаCEB)
тИаB +тИаCBE = 180┬░ ( As Linear pair)
тЗТтИаA = тИаB
(ii) тИаA+тИаD = тИаB+тИаC =180┬░ (Angles on the same side of transversal)
тЗТтИаA+тИаD = тИаA+тИаC (тИаA = тИаB)
тЗТтИаD = тИаC
(iii) In ╬ФABC and ╬ФBAD,
AB = AB (Common)
тИаDBA = тИаCBA
AD = BC (Given)
, ╬ФABC тЙЕ ╬ФBAD[SAS congruency]
(iv)Diagonal AC = diagonal BD by CPCT as ╬ФABC тЙЕ╬ФBA.