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M21-03

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M21-03 [#permalink]

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If in triangle \(ABC\) point \(D\) lies on side \(AC\), what is the value of \(\angle BCA\) ?


(1) The value of \(\angle ABD\) is half that of \(\angle BDC\)

(2) The value of \(\angle DBC\) is 10 degrees
[Reveal] Spoiler: OA

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Official Solution:

Statements (1) and (2) combined are insufficient. Let \(\angle ABD\) be \(x\). Then, \(\angle BDA = 180 - 2x\) and \(\angle BAC = x\). The sum of the angles of triangle ABC must equal 180. So, \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + (x) = 180\). Without knowing \(x\) we cannot find \(\angle BCA\).

Answer: E
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Re: M21-03 [#permalink]

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New post 25 Jul 2016, 08:20
how did you determine that angle ABD = angle BAC ?

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Re: M21-03 [#permalink]

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New post 05 Sep 2016, 20:59
aramesh19 wrote:
how did you determine that angle ABD = angle BAC ?

Because Angle BDC is an exterior angle to Angle BAC and Angle ABD

An exterior angle of a triangle is equal to the sum of the opposite interior angles

http://www.mathopenref.com/triangleextangletheorem.html
http://www.mathopenref.com/triangleextangle.html
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New post 25 Jan 2017, 11:07
why it is not C? Isn't BAC = 180 - 2x - 10?

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New post 26 Jan 2017, 06:06
bazu wrote:
why it is not C? Isn't BAC = 180 - 2x - 10?


When a DS question asks about the value of some variable, then the statement(s) is sufficient ONLY if you can get the single numerical value of this variable. Without knowing x we cannot get the measure of angle BAC.
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Re: M21-03 [#permalink]

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New post 07 Nov 2017, 04:36
The key here is that all DS questions need a definite answer. Solve for both the conditions one by one. Solve and you will get the answer as option E.
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New post 09 Nov 2017, 09:07
spetznaz wrote:
The key here is that all DS questions need a definite answer. Solve for both the conditions one by one. Solve and you will get the answer as option E.

There is no options for Math question, at least they are not published.

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New post 09 Nov 2017, 09:31
Bunuel wrote:
Official Solution:

Statements (1) and (2) combined are insufficient. Let \(\angle ABD\) be \(x\). Then, \(\angle BDA = 180 - 2x\) and \(\angle BAC = x\). The sum of the angles of triangle ABC must equal 180. So, \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + (x) = 180\). Without knowing \(x\) we cannot find \(\angle BCA\).

Answer: E

What the hell is answer E? Here is the solution:
(1) triangle ABD: \(\angle ABD + \angle BDA + \angle DAB = (x) + (180 - 2x) + (y) = 180\). From it we get \(x = y (\angle ABD = \angle DAB\)) or \(4x = 180\)
(2) Using where you ended up \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + (x) = 180\) we get:
Correct Answer - 80 - (click and drag your mouse to see the answer)

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New post 09 Nov 2017, 09:45
olexi wrote:
Bunuel wrote:
Official Solution:

Statements (1) and (2) combined are insufficient. Let \(\angle ABD\) be \(x\). Then, \(\angle BDA = 180 - 2x\) and \(\angle BAC = x\). The sum of the angles of triangle ABC must equal 180. So, \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + (x) = 180\). Without knowing \(x\) we cannot find \(\angle BCA\).

Answer: E

What the hell is answer E? Here is the solution:
(1) triangle ABD: \(\angle ABD + \angle BDA + \angle DAB = (x) + (180 - 2x) + (y) = 180\).From it we get \(x = y (\angle ABD = \angle DAB\)) or \(4x = 180\).
(2) Using where you ended up \(\angle BCA + \angle ABC + \angle BAC = \angle BCA + (x + 10) + (x) = 180\) we get:
Correct Answer - 80 - (click and drag your mouse to see the answer)


You are wrong. The correct answer is E. The highlighted part is not correct. From (x) + (180 - 2x) + (x) = 180 you don't get 4x = 180.
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New post 16 Nov 2017, 11:49
Bunuel wrote:
You are wrong. The correct answer is E. The highlighted part is not correct. From (x) + (180 - 2x) + (x) = 180 you don't get 4x = 180.


You are right, but I'm still not sure what answer E means.

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New post 16 Nov 2017, 19:40
olexi wrote:
Bunuel wrote:
You are wrong. The correct answer is E. The highlighted part is not correct. From (x) + (180 - 2x) + (x) = 180 you don't get 4x = 180.


You are right, but I'm still not sure what answer E means.


E means that Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

For this particular question it means that the measure of angle BCA cannot be determined, different values of x give different value of angle BCA.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: M21-03   [#permalink] 16 Nov 2017, 19:40
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