At what angle do the angle bisectors of two adjacent angles meet in a parallelogram?

(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