In the figure shown, is l1||l2 ?

Question

https://gmatclub.com/forum/download/file.php?id=54103 In the figure shown, is \ell_1 \parallel \ell_2 ? (1) r = s (2) t = u DS48710.02 2020-04-28_1909.png

BrentGMATPrepNow · Accepted Answer

Target question :    In the figure shown, is  l_1||l_2  ?

When I SCAN the two statements, I can see that statement 1 tells me about 2 angles created by the intersection of  line_1  and line m
And statement 2 tells me about 2 angles created by the intersection of  line_2  and line m
Since none of the information relates  line_1  and  line_2 , I’m pretty sure I can identify some cases with conflicting answers to the  target question . 
So, I’m going to head straight to……

Statements 1 and 2 combined   
  Key concept: For ANY 2 intersecting lines, the opposite angles will ALWAYS be equal

There are several scenarios that satisfy BOTH statements. Here are two: 
 Case a :
 https://i.imgur.com/Y686VKB.png 
In this case, the answer to the target question is  YES, of the two lines ARE parallel

Case b :
 https://i.imgur.com/UyZ2pgu.png 
In this case, the answer to the target question is  NO, of the two lines are NOT parallel

Since we cannot answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

RELATED VIDEO 
 https://www.youtube.com/watch?v=F26uam3kmCU

Kinshook · Answer

Asked: In the figure shown, is  l_1||l_2  ?

(1) r = s
This is possible for any intersection of 2 lines. r & s are opposite angles.
NOT SUFFICIENT

(2) t = u
This is possible for any intersection of 2 lines. t & u are opposite angles.
NOT SUFFICIENT

(1) + (2) 
(1) r = s
This is possible for any intersection of 2 lines. r & s are opposite angles.
(2) t = u
This is possible for any intersection of 2 lines. t & u are opposite angles.
Since there is nothing common between 2 statements
NOT SUFFICIENT

IMO E

MathRevolution · Answer

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. Visit https://www.mathrevolution.com/gmat/lesson for details. 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.

CrackverbalGMAT · Answer

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

GMATinsight · Answer

Answer: Option E

Video solution by  GMATinsight

https://www.youtube.com/watch?v=a8-UvxiiXjk

Kimberly77 · Answer

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

egmat · Answer

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:

https://gmatclub.com/forum/download/file.php?id=54103

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:

https://i.imgur.com/78TMPZz.png

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

Kimberly77 · Answer

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

In the figure shown, is l1||l2 ?

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In the figure shown, is l1||l2 ?

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