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A. neither equal nor supplementary

B. not equal but supplementary

C. equal but not supplementary

D. both equal and supplementary

Answer

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In our question, 2 configurations are possible. They are

Let us consider the configuration-1,

According to the question let us consider the lines AC is parallel to DM and AB is parallel to DF. We have to compare the angles, $ \angle CAB $ and $ \angle MDF $ .

In the figure, NO is parallel to PQ, the angle between the lines SR and NO is equal to the angle between the lines SR and PQ. This is called the corresponding angles property.

Using this property, from configuration-1 , we can write

AB is parallel to DF and MD acts as the transversal. So

$ \angle MEB=\angle MDF $

Similarly, the lines AC and MD are parallel to each other. So

$ \angle CAB=\angle MEB $

Using these two relations, we can write

\[\angle MDF=\angle MEB=\angle CAB\]

So we can conclude that the two required angles $ \angle CAB $ and $ \angle MDF $ are equal.

In the figure, NO is parallel to PQ, we can infer from the diagram that

$ \angle OTR+\angle SUQ=\pi $

For configuration-2, according to the question let us consider the lines AC is parallel to DM and AB is parallel to DF. We have to compare the angles, $ \angle IJK $ and $ \angle GHI $ .

For configuration-2 using the above property, we can write

GH is parallel to LI and GI is the transversal , so

$ \angle GHI+\angle LIH=\pi \to \left( 1 \right) $

From the corresponding angles property, we know that

$ \angle LIH=\angle IJK $ .

Using this result in equation-1 we get

$ \angle GHI+\angle IJK=\pi $

So, the two required angles are supplementary.

$ \therefore $ The required two angles can either be equal or supplementary depending on the configuration of the lines.