Question

A transversal intersects two parallel linee. prove that the bisectors of any pair of corresponding angles so formed are parallel ​

Answer 1

Answer:

:Let AB ║ CD and EF be the transversal passing through the two parallel lines at P and Q respectively. PR and QS are the bisectors of ∠EPB and ∠PQD.

 Since the corresponding angles of parallrl lines are equal,

∴∠EPB = ∠PQD

∴1/2 ∠EPB = 1/2 ∠PQD

∴∠EPR = ∠PQS

 But they are corresponding angles of PR and QS

Since the corresponding angles are equal

∴ PR ║ QS

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Let AB ║ CD and EF be the transversal passing through the two parallel lines at P and Q respectively. PR and QS are the bisectors of ∠EPB and ∠PQD.

 Since the corresponding angles of parallrl lines are equal,

∴∠EPB = ∠PQD

∴1/2 ∠EPB = 1/2 ∠PQD

∴∠EPR = ∠PQS

 But they are corresponding angles of PR and QS

Since the corresponding angles are equal

∴ PR ║ QS

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Answer 2

Step-by-step explanation:

The transversal AD intersects the two lines PQ and RS at points B and C respectively. BE is the bisector of ∠ABQ and CF is the bisector of ∠BCS.

As, BE is the bisector of ∠ABQ, then,

∠ABE=  

∠ABQ

In the same way,

∠BCF=  $\frac{1}{2}$∠BCS  

Since BE and CF are parallel and AD is the transversal, therefore, by corresponding angle axiom,

∠ABE=∠BCF

$\frac{1}{2}$∠ABQ=  

∠ABQ=∠BCS

Therefore, by the converse of corresponding angle axiom,

PQ∥RS.