In below figure PQ and RS are two equal and parallel line - segments. Any point M not lying on PQ or RS s joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that line segment MN and PQ are equal and parallel to each other

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Solution

$∵ P Q = R S a n d P N ∥ Q M \Rightarrow ∠ R P N = ∠ S Q M ⟶ (1) a n d P R = Q S ⟶ (2) a l s o, R N ∥ S M \Rightarrow ∠ P R N = ∠ Q S M ⟶ (3) T h e n, △ P R N ≅ △ Q S M (b y A S A) \Rightarrow P N = Q M a n d R N = S M (C P C T) s i n c e, P N ∥ Q M a n d P N = Q M \Rightarrow P Q M N i s a p a r a l l e l o g r a m s i m i l a r l y, R N ∥ S M a n d R N = S M \Rightarrow R S M N i s a p a r a l l e l o g r a m T h u s, P Q = M N a n d P Q ∥ M N$

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PQ and RS are two equal and parallel line segments.
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In below figure PQ and RS are two equal and parallel line - segments. Any point M not lying on PQ or RS s joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove that line segment MN and PQ are equal and parallel to each other

∵PQ=RSandPN∥QM⇒∠RPN=∠SQM⟶(1)andPR=QS⟶(2)also,RN∥SM⇒∠PRN=∠QSM⟶(3)Then,△PRN≅△QSM(byASA)⇒PN=QMandRN=SM(CPCT)since,PN∥QMandPN=QM⇒PQMNisaparallelogramsimilarly,RN∥SMandRN=SM⇒RSMNisaparallelogramThus,PQ=MNandPQ∥MN