Question

if a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel​

Answer 1

❄️ Solution:-

In the given figure in the attachment above, a transversal AD intersects two lines PQ and RS at points B and C respectively. Ray BE is the bisector of ∠ABQ and ray CG is the bisector of ∠BCS; and BE || CG.

We are to prove that PQ || RS.

It's given that the ray BE is the bisector of ∠ABQ.

Therefore, ∠ABE = 1/2 ∠ABQ (1)

Similarly, ray CG is the bisector of ∠BCS.

Therefore, ∠BCG = 1/2 ∠BCS (2)

But BE || CG and AD is the transversal.

Therefore, ∠ABE = ∠BCG

(Corresponding angles axiom) (3)

Substituting (1) and (2) in (3), we get

1/2 ∠ABQ = 1/2 ∠BCS

This is, ∠ABQ = ∠BCS

But, they are the corresponding angles formed by transversal AD with PQ and RS; and are equal.

Therefore, PQ || RS

(Converse of corresponding angles axiom)

Answer 2

Question:

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are pparallel.

Given:

• the transversal EH intersects the two lines AB and CD at points M and N respectively.
• MF is the bisector of $\tt\[\angle EMB\]$and NG is the bisector of $\tt\[\angle MND\]$.
• MF and NG are parallel to each other.

To prove:

We have to prove that lines AB and CD are parallel. Here, as MF is the bisector of

Solution:

$\tt\[\angle EMB\]$, so we get,$\tt\[\angle EMF=\dfrac{1}{2}\angle EMB = \left( i \right)\]$

Also, NG is the bisector of $\tt\[\angle MND\]$, so we get,$\tt\[\angle MNG=\dfrac{1}{2}\angle MND = \left( ii \right)\]$

Now by corresponding angle theorem,

we know that, if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

So here, since MF and NG are parallel and EH is the transversal,

∴by corresponding angle theorem, we get,$\tt\[\angle EMF=\angle MNG\]$

By substituting the values of $\sf\[\angle EMF\] \: nd \: \[\angle MNG\]$from equation (i) and (ii), we get,

Now by converse corresponding angle theorem, we know that, if two lines and a transversal form congruent corresponding angles, then the lines are parallel. So here, since corresponding angles

$\sf\[\angle EMB\] \: and \: \[\angle MND\]$are equal.

So we get,$\[AB||CD\]$

Hence proved.