kryton1212
A student claims that if one of the interior angles of a parallelogram is 30, then it cannot be inscribed in a circle. Do you agree with him? Explain.



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ash2326
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We start with assuming that the parallelogram is inscribed in a circle.
dw:1358656384414:dw
Assume this to be a parallelogram ABCD

kryton1212
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yes

ash2326
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dw:1358656432945:dw
Now angle A= angle C

kryton1212
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why they are equal?

ash2326
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A= C ( opposite angles of a parallelogram are equal)

kryton1212
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isn't that A+C=180?

kryton1212
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ohhh. parallelogram .... nothing then
continue please

ash2326
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For a cyclic quadrilateral opposite angles sum should be 180, we have to satisfy this property also

kryton1212
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ok...

ash2326
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so we have to have
\[A=C=90\]
this makes the parallelogram a rectangle, so any parallelogram which is inscribed in a circle is a rectangle

kryton1212
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and then>

ash2326
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Therefore a parallelogram with one angle 30 can't be inscribed

kryton1212
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ok...how about one angle is 150 degrees? the opp. angle can be 30

ash2326
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But opposite angles of a parallelogram are equal,

kryton1212
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ohhh yes..

kryton1212
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thank you so much

ash2326
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Welcome :D