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Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4476
Re: In the figure above, what is the value of x + y ?  [#permalink]

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russ9 wrote:
Hi Mike,

Thanks for the detailed response -- makes a lot of sense.

One point i'd like to bring up because I think that my original question wasn't phrased accurately:

1) If they give us that the height of ADC is 1/2 the height of ABC, doesn't that imply that the angle is doubled since AC is the same for both triangles? I meant to ask this and not bring in the "inscribed triangle" rule Thanks!

Dear russ9
Absolutely and 100% NO!! I don't know how to be any more clear about this. You are laboring under a very powerful geometry misconception. There is no easy proportional relationship between distances and angles. If it were, trigonometry would be such a ridiculous easy topic that teachers could cover it entirely in a day toward the end of a geometry class. Instead, trigonometry is such a difficult topic that it requires almost a full year on its own. If you have never learned trigonometry, suffice to say there are all kinds of subtleties about the relationship of lengths and angles that you do not and cannot appreciate. That's OK. You don't need to know any of that for the GMAT. All you have to do is rid yourself of this pernicious misconception.

Consider this diagram:
Attachment: Isosceles with half altitude.JPG [ 21.61 KiB | Viewed 3176 times ]

Triangle ABC is a 70-55-55 isosceles triangle. Point D is the midpoint of altitude BE, and the angle at D is not a nice even 140 degrees. Instead, it's an irrational angle, that is, an angle whose measure is an irrational number. This is very typically the case in trigonometry: for example, triangles with all integer sides must have at least two angles that have irrational measures, and triangles with integer angle measures almost always have at least two sides that have irrational lengths. Again, all this is much more than you need to know. I simply want to impress upon you that there are absolutely no magic proportions that link the behavior of lengths and the behavior of angles.

Does all this make sense?
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Manager  Joined: 15 Aug 2013
Posts: 226
Re: In the figure above, what is the value of x + y ?  [#permalink]

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mikemcgarry wrote:
russ9 wrote:
Hi Mike,

Thanks for the detailed response -- makes a lot of sense.

One point i'd like to bring up because I think that my original question wasn't phrased accurately:

1) If they give us that the height of ADC is 1/2 the height of ABC, doesn't that imply that the angle is doubled since AC is the same for both triangles? I meant to ask this and not bring in the "inscribed triangle" rule Thanks!

Dear russ9
Absolutely and 100% NO!! I don't know how to be any more clear about this. You are laboring under a very powerful geometry misconception. There is no easy proportional relationship between distances and angles. If it were, trigonometry would be such a ridiculous easy topic that teachers could cover it entirely in a day toward the end of a geometry class. Instead, trigonometry is such a difficult topic that it requires almost a full year on its own. If you have never learned trigonometry, suffice to say there are all kinds of subtleties about the relationship of lengths and angles that you do not and cannot appreciate. That's OK. You don't need to know any of that for the GMAT. All you have to do is rid yourself of this pernicious misconception.

Consider this diagram:
Attachment:
Isosceles with half altitude.JPG

Triangle ABC is a 70-55-55 isosceles triangle. Point D is the midpoint of altitude BE, and the angle at D is not a nice even 140 degrees. Instead, it's an irrational angle, that is, an angle whose measure is an irrational number. This is very typically the case in trigonometry: for example, triangles with all integer sides must have at least two angles that have irrational measures, and triangles with integer angle measures almost always have at least two sides that have irrational lengths. Again, all this is much more than you need to know. I simply want to impress upon you that there are absolutely no magic proportions that link the behavior of lengths and the behavior of angles.

Does all this make sense?
Mike Thanks, Mike. Make's total sense. Appreciate the help.
Intern  Joined: 03 Sep 2013
Posts: 1
Re: In the figure above, what is the value of x + y ?  [#permalink]

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Hi Bunuel,
So this means the OG explanation is wrong (the explanation, not the answer)?
OG says: "If ΔABC and ΔADC are isosceles triangles, then ∠BAC and ∠BCA have the same measure, and ∠DAC and ∠DCA have the same measure. However, since no values are given for any of the angles, there is no way to evaluate x + y; NOT sufficient."
From my understanding OG can't assume ∠BAC and ∠BCA are the same. That's not stated in any part of the problem. It just says it's an isosceles triangle. We don't know what sides are equal. Same for ΔADC even if the figure let you to believe AD and DC are equal.
Am I correct?

Gustavo
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Re: In the figure above, what is the value of x + y ?  [#permalink]

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Bunuel wrote: In the figure above, what is the value of x + y ?

(1) x = 70.
(2) ABC and ADC are both isosceles triangles.

Even when we consider both statements together we don't know the placement of points D and B. For example consider the diagram below:
Attachment:
The attachment Triangles2.png is no longer available
As you can see we can have two different answers for x+y for two different placements of point D.

The only issue I see in this poblem, that we don't know by St1+St2) whether two other angles besides X are equal, or there is X and one other angle, that are equal. But I don't understand the point with D, what do you mean with two different placements of point D.

Do you mean what I wrote below but algebraically ?
Let's say two other angles besides X are equal, the we could write down some equations:
Y+2Z=180°
X+2Z+2W=180 --> 2Z+2W=180-70=110 --> Z+W=55. But we nees to find a value for Z in order to find Y. So this combined 55° don't help us....

Update

(1) x=70° clearly not sufficient
(2) Both triangles are isoceles, ok, but we don't know which two sides of those triangles are equal.. means we get different angles Not sufficient
(1)&(2) is still not sufficient because of points stated in (2)
Dear Experts, in statement 2 we can not just assume which of the sides are equal, there is no information stated in the question..Or I'm wrong....?
Attachments 149.png [ 7.29 KiB | Viewed 2667 times ]

Director  G
Joined: 02 Sep 2016
Posts: 638
Re: In the figure above, what is the value of x + y ?  [#permalink]

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mike and Bunuel

Is this reasoning correct?

To find: x+y
1) x=70

y=?

2) They are isosceles triangle but we have no idea which two sides are equal and thus which two angles are equal.

Both
Can't really find a unique value here.

E
Director  G
Joined: 02 Sep 2016
Posts: 638
Re: In the figure above, what is the value of x + y ?  [#permalink]

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Also Bunuel and mike

Why are you considering the alignment of two angles x and y ?
Is there some rule ?

Thanks
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4214
In the figure above, what is the value of x + y ?  [#permalink]

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Top Contributor
Attachment:
Triangles.png
In the figure above, what is the value of x + y ?

(1) x = 70
(2) ABC and ADC are both isosceles triangles

Target question: What is the value of x + y ?

IMPORTANT: For geometry Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the video below.

When I use the above strategy, I can see that neither statement (alone) LOCKS in the value of y.
Given this, I can jump straight to...

Statements 1 and 2 combined
Statement 1 LOCKS in the value of x
Statement 2 tells us that ∆ABC and ∆ADC are isosceles triangles, but this is not enough to lock in the value of y.
Consider the following two diagrams that satisfy BOTH statements:

DIAGRAM #1 IMPORTANT: Notice that I can mentally take point D and push it down to get...
DIAGRAM #2 Notice that, by mentally pushing point D down, we don't change the fact that x = 70 degrees, AND we don't change the fact that ∆ABC and ∆ADC are isosceles triangles
However, when we mentally push point D down, we CHANGE the value of y

Since the value of y can vary, the value of x+y will also vary.
In other words, we cannot answer answer the target question with certainty.
As such, the combined statements are NOT SUFFICIENT

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Re: In the figure above, what is the value of x + y ?  [#permalink]

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