In the given figure, ABCD is a quadrilateral with ∠1=∠2 and ∠3=∠ABC and ∠4=∠BAD. Also, AP and BQ are produced to points E and F, respectively. Then, we can say that
A
ΔOFE is an equilateral triangle
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B
ΔOFE is an isosceles triangle
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C
ΔOFE is a scalene triangle
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D
Can't be determined
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Solution
The correct option is BΔOFE is an isosceles triangle Given, ABCD is a quadrilateral with ∠1=∠2,∠3=∠ABC and ∠4=∠BAD In ΔAPB and ΔEPC, ∠APB=∠CPE [vertically opposite angles] ∠PCE=∠3=∠PBA [given] ∴ΔAPB∼ΔEPC [by AA similarity] ⇒∠PEC=∠1 ...(i) Similarly, in ΔFQD and ΔBQA, ∠QFD=∠2 ...(ii) From Eqs. (i) and (ii), we get ∠PEC=∠QFD[∵∠1=∠2,given] ⇒∠OFE=∠OEF ⇒OF=OE ∴ΔOFE is an isosceles triangle.