Question

abcd is a quadrilateral in which ab=ad and bc=dc. diagonals ac and bd intersect at o show that triangle abc is congruent to triangle adc ii)triangle aob is congruent to triangle aod iii) ac is perpendicular to bd

Answer 1

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Answer 2

(1)  Note that in △ABC and △ADC, we have    AB=CD    (given)   AD=BC    (given)  AC=AC     (common side)Thus it impplies that,   △ABC≅△ADC   (By SSS congruency)By CPCT, we get     ∠DAC=∠BACor  ∠DAO=∠BAO    ....(1)   (since AOC is a staright line)(2)  Note that in △AOB and △AOD, we have          AD=BC           (given)           AO=AO     (common side)  ∠DAO=∠BAO     (from (1))Thus it impplies that,   △AOB≅△AOD  (By ASA congruency)(3)  In previous part,we have proved that   △AOB≅△AODThen by CPCT, we get    ∠AOD=∠AOB    ....(2)But these form linear pair, so we get     ∠AOD+∠AOB=180°