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Question
In the adjacent figure, if seg AB || seg PQ, seg AB ≅ seg PQ, seg AC || seg PR, seg AC ≅ seg PR then prove that, seg BC || seg QR and seg BC ≅ seg QR.
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Solution
Given: seg AB || seg PQ, seg AB ≅ seg PQ,
seg AC || seg PR, seg AC ≅ seg PR
To prove: seg BC || seg QR, seg BC ≅ seg QR
Proof:
In `square`ABQP,
seg AB || seg PQ
seg AB ≅ seg PQ ...(Given)
∴ `square`ABQP is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ segAP || segBQ ...(i)
∴ seg AP ≅ seg BQ ...(ii) ...(Opposite sides of a parallelogram)
In `square`ACRP,
seg AC || seg PR
seg AC ≅ seg PR ...(Given)
∴ `square`ACRP is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ seg AP || seg CR ...(iii)
∴ seg AP ≅ seg CR ...(iv) ...(Opposite sides of a parallelogram)
In `square`BQRC,
seg BQ || seg CR ...[From (i) and (iii)]
seg BQ ≅ seg CR ...[From (ii) and (iv)]
∴ `square`BQRC is a parallelogram. ...(A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and congruent)
∴ seg BC || seg QR
∴ seg BC ≅ seg QR ...(Opposite sides of parallelogram)
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