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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.

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.

And AC = AC   [common side]

∴∠B=∠D    [SSS theorem] ————————- (i)

Again, in ∆ABD and ∆BCD,

And BD=BD    [common side]

∴∆ABD≌∆BCD

∴∠A=∠C   [SSS theorem] ——————————-(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.

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.

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)

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...