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Home / Geometry / Prove that, if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram

Prove that, if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram

Exercise: Prove that, if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

General enunciation: We have to prove that if the opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.

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Particular enunciation: Let the opposite sides of the quadrilateral ABCD are equal and parallel .i.e. BC=AD,BC∥AD and AB=DC,AB∥DC

We have to prove that ABCD is a parallelogram.

Construction: We join A,C and B,D.

Proof: In ∆ABC and ∆ADC,

AB=DC, BC=AD   [supposition]

And AC = AC   [common side]

∴∆ABC≌∆ADC

∴∠B=∠D    [SSS theorem] ————————- (i)

Again, in ∆ABD and ∆BCD,

AB=DC, BC=AD      [supposition]

And BD=BD    [common side]

∴∆ABD≌∆BCD

∴∠A=∠C   [SSS theorem] ——————————-(ii)

Now, in the quadrilateral ABCD,

BC=AD,AB=DC           [from (i)and (ii)]

∴ ∠A =C, ∠B=∠D [∵the opposite sides and angles of a parallelogram are equal]

∴ABCD is a parallelogram    [Proved]

Exercise: Prove that, the bisectors of any two opposite angles of a parallelogram are parallel to each other.

General enunciation: To prove that the bisectors of any two opposite angles of a parallelogram are parallel to each other.

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Particular enunciation: Let ABCD is a parallelogram.

It’s the bisectors of any two opposite angles ∠ABC and ∠ADC are BF and DF respectively.

We have to prove that DE∥BF.

Proof: ABCD is a parallelogram.

So ∠ABC=∠ADC        [∵opposite angles of a parallelogram are equal]

Since BF and DE are their bisectors respectively.

∴∠EBF=∠EDF              ————————————— (i)

Again, AB∥CD and BF is their transversal

∴∠EBF=∠BFC               ————————- (ii) [alternative angles are equal]

Now, ∠EDF=∠EBF=∠BFC

∴∠EDF=∠BFC         [from (i) and (ii)]

But these are corresponding angles where DF is the transversal.

Therefore DE∥BF    (Proved)

Exercise: Prove that, the bisectors of any two adjacent angles of a parallelogram are perpendicular to each other.

General enunciation: To prove that the bisectors of any two adjacent angles of a parallelogram are perpendicular to each other.

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Particular enunciation: Let ABCD is a parallelogram. It’s the bisectors of two adjacent angles ∠ABC and ∠BCD are BO and CO respectively.

We have to prove that BO⊥CO.

Proof: ABCD is a parallelogram so AB∥CD and BC is their transversal

When a transversal cuts two parallel straight lines then the sum of the pair of interior angle on the same side of the transversal is 2 right angles.

∴∠ABC+∠BCD=1800                              

Or, ½  ∠ABC + ½ ∠BCD=900

Or, ∠OBC+∠OCB = 900       —————————- (1)

In ∆BOC, we know that sum of three angles of a triangle is equal to 2 right angles

∴ ∠OBC+∠OCB+∠BOC=1800                         

Or, 900+∠BOC=1800       [from (1)]

Or, ∠BOC=1800-900

Or, ∠BOC=900

Therefore BO⊥CO   (Proved)

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