(i) Given AM bisect angle A and BM bisects angle of || gm ABCD.
Hence, bisectors of any two adjacent angles of a parallelogram are at right angles.
(ii) Given: A || gm ABCD in which bisector AR of ∠A meets DC in R and bisector CQ of ∠C meets AB
in Q
To prove: AR || CQ
Proof:
In || gm ABCD, we have
∠A = ∠C [Opposite angles of || gm are equal]
½ ∠A = ½ ∠C
∠DAR = ∠BCQ [Since, AR is bisector of ½ ∠A and CQ
is the bisector of ½ ∠C]
Now, in ∆ADR and ∆CBQ
∠DAR = ∠BCQ [Proved above]
AD = BC [Opposite sides of || gm ABCD are
equal]
So, ∆ADR ≅ ∆CBQ, by A.S.A axiom of congruency
Then by C.P.C.T, we have
∠DRA = ∠BCQ
And,
∠DRA = ∠RAQ [Alternate angles since, DC || AB]
Thus, ∠RAQ = ∠BCQ
But these are corresponding angles,
Hence, AR || CQ.
(iii) Given: In quadrilateral ABCD, diagonals AC and BD are equal and bisect each other at right angles
To prove: ABCD is a square
Proof:
In ∆AOB and ∆COD, we have
AO = OC [Given]
BO = OD [Given]
∠AOB = ∠COD [Vertically opposite angles]
So, ∆AOB ≅ ∆COD, by S.A.S axiom of congruency
By C.P.C.T, we have
AB = CD
and ∠OAB = ∠OCD
But these are alternate angles
AB || CD
Thus, ABCD is a parallelogram
In a parallelogram, the diagonal bisect each other and are equal
Hence, ABCD is a square.
In ∆ABC and ∆DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, Band C are joined to vertices D, E and F respectively (see figure). 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. [CBSE 2012
Given: 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.To Prove: (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.Proof: (i) In quadrilateral ABED,AB = DE and AB || DE| Given∴ quadrilateral ABED is a parallelogram.| ∵ A quadrilateral is a parallelogram if a pair of opposite sides are paralleland are of equal length(ii) In quadrilateral BEFC,BC = EF and BC || EF | Given∴ quadrilateral BEFC is a parallelogram.| ∵ A quadrilateral is a parallelogram if a pair of opposite sides are paralleland are of equal length(iii) ∵ ABED is a parallelogram| Proved in (i)∴ AD || BE and AD = BE ...(1)| ∵ Opposite sides of a || gmare parallel and equal∵ BEFC is a parallelogram | Proved in (ii)∴ BE || CF and BE = CF ...(2)| ∵ Opposite sides of a || gmare parallel and equalFrom (1) and (2), we obtainAD || CF and AD = CF.(iv) In quadrilateral ACFD,AD || CF and AD = CF| From (iii)∴ quadrilateral ACFD is a parallelogram.| ∵ A quadrilateral is a parallelogram if a pair of opposite sides are parallel and are of equal length(v) ∵ ACFD is a parallelogram| Proved in (iv)∴ AC || DF and AC = DF.| In a parallelogram opposite sides are parallel and of equal length(vi) In ∆ABC and ∆DEF,AB = DE| ∵ ABED is a parallelogramBC = EF| ∵ BEFC is a parallelogramAC = DF | Proved in (v)∴ ∆ABC ≅ ∆DEF.
| SSS Congruence Rule