Opposite angles in a parallelogram, why are congruent.

To prove is a diagonal of parallelogram.

Class 9th Maths, Page No:147, Questions No:9, Exercise 8.1

NCERT, Solutions for Class 9th Maths, Session 2023-2024.

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(i) In ΔAPD and ΔCQB,

DP = BQ [∵ Given]

∠ADP = ∠CBQ [∵ Alternate angle]

AD = BC [∵ Opposite sides of a parallelogram]

Hence, ΔAPD ≅ ΔCQB [∵ SAS Congruency rule]

(ii) ΔAPD ≅ CQB [∵ Prove above]

AP = CQ …(1) [∵ CPCT]

(iii) In ΔAQB and ΔCPD,

QB = DB [∵ Given]

∠ABQ = ∠CDP [∵ Alternate angle]

AB = CD [∵ Opposite sides of a parallelogram]

Hence, ΔAQB ≅ ΔCPD [ SAS Congruency rule]

(iv) ΔAQB ≅ ΔCPD [∵ Prove above]

AQ = CP …(2) [∵ CPCT]

(v) In APCQ,

AP = CQ [∵ From (1)]

AQ = CP [∵ From (2)]

The opposite sides of quadrilateral APCQ are equal.

Hence, APCQ is a parallelogram.