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Is the measure of one of the interior angles of

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Is the measure of one of the interior angles of

by BTGmoderatorLU » Tue Aug 28, 2018 2:54 pm

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Source: GMAT Prep

Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

(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.

The OA is E.
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Source: — Data Sufficiency |
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by Brent@GMATPrepNow » Tue Aug 28, 2018 3:05 pm

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BTGmoderatorLU wrote:Source: GMAT Prep

Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

(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.

The OA is E.
Target question: Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

Key concept: the 4 angles in a quadrilateral must add to 360 degrees

Statement 1: Two of the interior angles of ABCD are right angles.
Let's test some possible cases.
There are infinitely many quadrilaterals that satisfy statement 1. Here are two:
Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is YES, one of the angles IS 60°
Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is NO, one of the angles is NOT 60°
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The degree measure of angle ABC is twice the degree measure of angle BCD.
Let's test some possible cases.
Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is YES, one of the angles IS 60°
Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is NO, one of the angles is NOT 60°
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is YES, one of the angles IS 60°
Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is NO, one of the angles is NOT 60°
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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Is the measure of one of the interior angles of

by fskilnik@GMATH » Wed Aug 29, 2018 9:29 am

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BTGmoderatorLU wrote:Source: GMAT Prep

Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?

(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.
This is a perfect opportunity to present one of GMATH´s most powerful tools in Geometry-related Data Sufficiency problems: the GEOMETRIC BIFURCATION

Obs.: all angles in the GMAT are in degrees, therefore this unit will be omitted in the wording and also in the figures shown!

? : (at least) one internal angle measures 60?

(1) We present:
> A square that answers in the negative
> A square-based-but-modified CONSTRUCTIBLE (*) figure that answers in the affirmative

(*) Possible to be done by straight-edge and compass. (This is important to VALIDATE the argument, it´s not a drawing anymore, it´s a geometrical counterexample to the "always NO" argument.)

(2) We present:
> A 45 and 90 consecutive internal angles and an easy CONSTRUCTIBLE "closing the polygon construction", answering NO.
> A 45 and 90 consecutive internal angles and an easy "modified-previous-drawing-closing-the-polygon" in a CONSTRUCTIBLE way, answering YES. (Think about 5 seconds to realize that, following the dotted ray from the previous construction.)

(1+2) We present:
> An easy CONSTRUCTIBLE figure, answering NO.
> The fact that it is not that easy to guarantee YES is possible, we use Algebra to help us... is the last figure shown geometrically possible? YES. (180+3x = 360 implies x=60 and the construction is possible... now it´s easy to realize that!)
Obs.: the vertices A, B, C, D are not in the same positions of the previous figures, but this doesn´t matter! (Statements (1) and (2) are still obeyed, and that is really crucial.)

Some considerations on the Geometric Bifurcation follows:

1. Could we try to bifurcate (1+2) right at first?
Sure, and if you succeed it would spare some (say) 20/30 seconds, what is always good.

2. Drawings are mathematically valid?
No, but CONSTRUCTIBLE drawings are!
The fact that most students are not used to geometric bifurcations its just a matter of ... not being exposed to it previously.
It´s mathematically rigorous. It´s quick after some practice. It makes your "intuition" shielded!

3. Why not simply avoid drawings?
Because you must guarantee that the numbers you present may be put in a geometric viable construction.

I hope you enjoy (and see the power) of this!

Regards,
fskilnik.


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