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Home / Geometry / The straight lines which are parallel to the same straight line are parallel to one another.

The straight lines which are parallel to the same straight line are parallel to one another.

Theorem: If a straight line intersects another two straight lines and if the corresponding angles are equal to each other.

General enunciation: If a straight line intersects another two straight lines and if the corresponding angles are equal to each other.

Particular enunciation: Let the straight line PQ intersects AB and CD at the points R and S respectively, so that it becomes exterior ∠PRB = interior opposite ∠RSD.

It is required to prove that, AB and CD are parallel.

Proof: Since the ∠ARS =∠PRB (being vertically opposite angles)

And ∠ PRB =∠RSD (Given)

∴∠ARS =∠RSD. But they are alternate angles.

∴AB and CD are parallel.                        (Proved

 

Theorem: The sum of the two interior angles in the same side of the bisector is equal to those two right angles then two lines are parallel.

General enunciation: The sum of the two interior angles in the same side of the bisector is equal to those two right angles then two lines are parallel.

 

Particular enunciation: Let the straight line EF intersects the straight lines   AB and CD at the points R and S respectively, so that in the some side of PQ,

interior ∠BRS + interior ∠RSD = two right angles

It is required to prove that, AB and CD are parallel.

Proof: Now, ∠ARS +∠BRS = two right angles

And ∠BRS + ∠RSD = two right angles (Given)

∴∠ARS +∠BRS =∠BRS + ∠RSD.

Now subtracting ∠BRS from both the sides we get, ∠ARS =∠RSD. But they are alternate angles.

∴AB and CD are parallel.                    (Proved).

 

 

Theorem: The straight lines which are parallel to the same straight line are parallel to one another.

General enunciation: The straight lines which are parallel to the same straight line are parallel to one another.

Particular Enunciation: Let the straight lines AB, CD be each parallel to the straight line EF.

It is required to prove that, AB and CD are parallel to one another.

Construction: Draw a straight line PQ cutting AB, CD and EF at the points X, Y and Z respectively.

Proof: Since AB and EF are parallel, PQ is their transversal, then

∠AXQ = the alternate ∠PZF

Again, Since CD and EF are parallel; PQ is their transversal, then

∠PYD = the corresponding ∠PZF

∴ ∠AXQ = ∠PYD.

∴ AB and CD are parallel to one another.                             (Proved)

 

 

For PDF File click Here: Theorem – parallel (class 8)

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