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Re: Common Mistakes in Geometry Questions - Exercise Question #2 [#permalink]

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20 Aug 2017, 09:33

Not sure why B is insufficient. According to statement 2, BC is a straight line and perpendicular to AD. It can be inferred that Angle BCD = Angle BDC = 45°

Statement 2 is not sufficient here. The solutions above that contend it is sufficient are all making assumptions that you don't know to be true. For example, you don't know that the vertical-looking line bisects the 90 degree angle - you'd need to know that the large triangle was isosceles (AC = CD) to conclude that, and we don't know whether AC = CD is true. Nor do we know that the vertical-looking line is perpendicular to the horizontal-looking line (i.e. we don't know that AD is perpendicular to BC). If you did know either of those things, Statement 2 would be sufficient, but we don't know those things.

You can see visually why Statement 2 is not sufficient. Draw a circle, with center B, and draw a diameter AD. Then let C be any other point on the circumference of the circle. Draw the triangle ADC. We must get a 90 degree angle at C, since we're connecting a diameter to a point on the circle, and every radius is the same length, so BC = BA = BD. By moving the point C around the circumference, you can easily change the size of the two smaller angles in triangle ADC (the angles at A and at D). Now just erase the circle and you have the same situation as is described in this question, when using Statement 2 alone. Since we just saw we can move point C around and change the size of angle CDB, the information can't be sufficient.
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Re: Common Mistakes in Geometry Questions - Exercise Question #2 [#permalink]

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14 Sep 2017, 19:36

Having a very hard time understanding any of the explanations of why statement 2 is insufficient. The question seems to imply that angle ACD is 90 degrees. If AB = BC = BD, doesn't that mean that angle ACB + BCD = 90?

Having a very hard time understanding any of the explanations of why statement 2 is insufficient. The question seems to imply that angle ACD is 90 degrees. If AB = BC = BD, doesn't that mean that angle ACB + BCD = 90?

Yes, angle ACD is 90 degrees and ACB + BCD = 90 degrees but the point is that we don't know whether CB is perpendicular to AD. Not knowing that, how are you getting that (2) is sufficient?
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