In the figure shown, is l1||l2 ?

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Condition 1)
Angles r and s are vertical angles and they are always equal each other whatever r and s are. It means condition 1) tells nothing.

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Angles t and u are vertical angles and they are always equal each other whatever t and u are. It means condition 1) tells nothing.

Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
Both conditions tell nothing.

Since both conditions together do not yield a unique solution, they are not sufficient.

Therefore, E is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

Solution:
A new og 2021 question

St(1):- r = s

Vertically opposite angles in any two intersecting lines are equal.

It does not give us enough information to determine if l1 || l2 (Insufficient)

St(2):-t = u

Vertically opposite angles in any two intersecting lines are equal.

It does not give us enough information two determine if l1 || l2 (Insufficient)

Combining both;

r=s and t = u are true for two intersecting lines (Vertically opposite angles in intersecting lines are equal)

However, no unique information can be deduced for angles s and t / r and u. (Insufficient) option (e)

Hope this helps
Devmitra Sen (GMAT Quant Expert

Answer: Option E

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Hi BrentGMATPrepNow, to clarify so opposite angles are always equeal regardless of the two lines are parallel or not.
So in order for two lines to parallel, does it mean that if 140+40 (within L1 & L2) = 180 ? or 40 above L1 = 40 above line 2? or what else? Thanks Brent

Hi Kimberly77
Thanks for your query.


Let me first answer your questions one by one.


Question 1- “So opposite angles are always equal regardless of whether the two lines are parallel or not?”

The first thing you need to understand is when are two angles called opposite angles. For a pair of opposite angles to form, we need TWO INTERSECTING LINES. This has got nothing to do with parallel lines. In fact, having two parallel lines means that they will never intersect and that opposite angles will never be formed. (I know you were using “parallel” only in a context similar to this question, but you still need to have your terminology set.)

In the given figure:

• \(l_1\) and m are two intersecting lines. Here, angles t and u are opposite angles and hence EQUAL as well.
  
  • Here, for t and u being equal, \(l_2\) has no role to play at all.
• Similarly, for \(l_2\) and m, s and r are opposite angles and hence, equal. Again, this is irrespective of the presence of \(l_1\).

Question 2- “for two lines to be parallel, does it mean that if 140+40 (within L1 & L2) = 180 ? or 40 above L1 = 40 above line 2? or what else?”

You are thinking correctly. To tell you “what else” properly, I will first redraw the figure with \(l_1, l_2\), and m. This time, I am also numbering all of the eight angles formed for easy reference:



Now, in this figure, \(l_1\) and \(l_2\) are two lines being cut by a single transversal, line m.

1. Here, we have 4 pairs of corresponding angles: 1 and 5, 4 and 8, 2 and 6, and 3 and 7.
   
   • Whenever two parallel lines are cut by a single transversal, these pairs of corresponding angles are equal. (1 = 5; 4 = 8; 2 = 6; 3 = 7).
   • Conversely, if corresponding angles are equal, then the two lines being cut by a single transversal are parallel.
2. The figure also shows two pairs of alternate interior angles: 4 and 6, 3 and 5.
   
   • Whenever two parallel lines are cut by a single transversal, these pairs of alternate interior angles are equal. (4 = 6; 3 = 5)
   • Conversely, if alternate interior angles are equal, then the two lines being cut by a single transversal are parallel.
3. Next, the figure shows two pairs of interior angles on the same side of the transversal (4 and 5, 3 and 6).
   
   • Whenever two parallel lines are cut by a single transversal, these pairs of interior angles on the same side of the transversal are supplementary (4 + 5 = 180°; 3 + 6 = 180°).
   • Conversely, if interior angles on the same side of transversal make supplementary pairs, then the two lines being cut by a single transversal are parallel.

Hope this will help!

Best,
Aditi Gupta
Quant expert, e-GMAT

Oh wow what a great explanation and all my confusions are clear now. Thank you so much @e-GMAT !!!