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DS - Geometry

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DS - Geometry

by mav800rick » Tue May 27, 2008 4:06 pm
Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees?
(1) Two of the interior angles of ABCD are right angles.
(2) The degree measure of angle ABC is twice the degree measure of angle BCD.

-Keen to see the approach used.
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by netigen » Tue May 27, 2008 8:03 pm
Any two angles are 90, we do not know which ones.
One angle is twice the other ABC = BCD

If none of the angles ABC and BCD are 90 then we can say yes
if BCD = 90 then we do not know, it may be or may not be
if ABC = 90 then we do not know, it may be or may not

hence ans will be E
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by shiraj » Wed May 28, 2008 12:34 am
Ans: E

(1) Two angles measure 90 degrees.
This only says the sum of the rest two angles measure 180 degree.

So option A, D is ruled out.

(2) Angle ABC = 2 (Angle BCD)

This condition alone cannot answer. So option B is ruled out.

Now we are left with 3 cases (as both the angles cannot be 90 degree)-
Case 1 - If Angle ABC=90, then Angle BCD = 45,
Another angle is 90, so the fourth angle wud be 135 degree.
So the answer to the question, whether or not any of the angle is 60 degree, is NO
Case 2 - If Angle BCD=90, then Angle ABC = 180,
Another angle is 90, that means the forth angle is of 0 degree,
which is not a possible solution.
Case 3 - If angles other than ABC , BCD are 90 degrees.
In this case, Angle ABC + Angle BCD = 180
=> x + 2x = 180 => x=60
So the answer to the question, whether or not any of the angle is 60 degree, is YES


Hence the answer, (E)
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OA is 'C'

by mav800rick » Wed May 28, 2008 6:16 am
Even I thought it is 'E'
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OA given in 'C'

by mav800rick » Wed May 28, 2008 6:55 am
[quote="netigen"]Any two angles are 90, we do not know which ones.
One angle is twice the other ABC = BCD

If none of the angles ABC and BCD are 90 then we can say yes
if BCD = 90 then we do not know, it may be or may not be
if ABC = 90 then we do not know, it may be or may not

hence ans will be E[/quote]
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by chidcguy » Wed May 28, 2008 10:29 am
Lets put them together

(1) Two of the interior angles of ABCD are right angles.
(2) The degree measure of angle ABC is twice the degree measure of angle BCD.

A+B+C+D=360

two of them are right angles and two angles are such that one angle is double the other. Is is possible that one of the interior angle which is 90 degrees has another angle which is 2 X 90 =180 degrees

Cannot be because we know that two angles are 90 degrees & if there is 180 degree angle fourth angle must be zero.

That means angle b = 2 times angle c
&
angle a, d are 90 degrees

a+b+c+d=360

and b+c=180 substituting b=2c we get 3c=180 and c = 60 degrees.

So we can say that Yes, one angle is 60 degrees

Is the OA c? If there are any mistakes, please point out.
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