The figure above shows a portion of a road map on which the measures

The Precise approach would start off using the equations a+b+c=180 and b=2a.This gives us: a+2a+c=180, which is not enough to find neither a nor c. Answer choices (A) and (D) are eliminated.
In the same way, if we take a+b+c=180 and c=3a,we'll get a+b+3a=180, which is not enough to find neither a nor b. Answer choice (B) is eliminated.
Using both statements we'll get a+2a+3a=180, so 6a=180, and thus a=30. Now, since c=90 the lines are indeed parallel (the sum of the two adjacent angles, both c, is 180). This means that both statements together are enough.

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If PQ and RS are two parallel lines, then the line across them is a transversal line, and so the sum of the two adjacent angels must be equal,
=> c=180–c
=> 2c=180
=>c=90
Thus, if c=90, PQ and RS must be parallel.

So, we have to check if c=90.

1) No way to find the value of c.
2) No way to find the value of c.
3) Using 1) and 2) we can find the value of c. Thus, C is the answer.

The Logical approach to this question would start with the fact that each of the triangles and the point in which they 'touch' (T) all provide the same equation: a+b+c=180. We'll use the Logical rule that when there are more variables than equations, there's an infinite number of possible solutions. Now, since each of the two statements only eliminates one of the variables, we'll need both of them to find the exact size of the angles.
Note: there's no need to find what this size is, since with the exact value we can definitely answer the question (either 'Yes!' or 'No!'), so there's enough information.

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We can prove two lines are parallel if we know Angle C i.e. if C=90 then parallel else No

1. b=2a. No info about c. Not Sufficient
2. c=3a. No infor about b. Not Sufficient

On combining, we can get values of all a,b and c
Sufficient.

Hence C

Are we supposed to assume for this question that PS is a straight line? I chose "E" because I didn't want to assume PS as a straight line and the similar triangles could potentially be rotated around point T making the lines potentially not parallel even if c=90 degrees.

OFFICIAL GUIDE:

Problem Solving
Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency:
Figures:
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.

Hope it helps.
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Ok, i am tipping over why we have to prove C=90 degrees? Slope of two lines need to be equal right for them to be parallel?

Draw a line (AB) parallel to RS passing through point T. now we need to show is PQ || to our construction. For this to be parallel, QT must be a transverse and alternate angles must be equal .i.e.

Angle PQT = Angle QTA
Angle PQT = Angle PTA - Angle QTP
which is :
b = c-a

Now with option A:
We are getting
2a = c-a
3a =c (NOT SUFFICIENT as we dont know about a & c)

With option B:
This gives the SAME RELATION AS ABOVE. Hence, together is SUFFICIENT (C)

Sir, could you please explain why c must equal 90 degree for the two lines to be parallel? I mean if say c= 89, both lines would be making the same angle from x axis and hence they should be parallel. Am I missing something? Plus, don't you think that since both the lines have c as angle with x axis, the question is already establishing that both lines are parallel since c can have only one value? Shouldn't the two angles be denoted with different variables in this question?

Along a transversal the angle has to be on the same side for both lines. Imagine the c degree being 45, in such a case both lines would intersect. If and only if they are 90 degrees they would not. Hence you try and find value of c

The figure above shows a portion of a road map on which the measures of certain angles are indicated. If all lines shown are straight and intersect as shown, is road PQ parallel to road RS ?

The two lines are parallel only if c = 90 degrees. Lets take a look at the statements:

(1) b = 2a

We can't conclude anything about c; INSUFFICIENT.

(2) c = 3a

We can't conclude anything about c; INSUFFICIENT.

(1&2) 3a + 2a + a = 180
a = 30; b = 90

SUFFICIENT.

Answer is C.
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Sure I understand C is the right answer, but as per the figure lines PQ and RS both intersect the road at angle C , as they both intersect the line with the same angle they are parallel . Isnt this enough ?

(1) It gives an equation:\(3a+c=180,\) still \(c\) is unknown; Insufficent.

(2) It gives an equation: \(4a+b=180\), still \(b\) is unknown; Insufficent.

Considering both:
\(3a+c=180,\) and \(c=3a\)

Then \(6a=180\): Sufficient

The answer is C
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Hello Bunuel

The angle C is the same at the points "P" and "S" (per figure), and since the line, "PS" is already straight (assumption) aren't the lines "QP" and "RS" going to be parallel, irrespective of the angle?
I am not sure why should we prove that angle C is equal to 90?

Let me ask you: say angle c is 30 degrees, would the lines be parallel in this case?
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Hello Bunuel

The angle C is the same at the points "P" and "S" (per figure), and since the line, "PS" is already straight (assumption) aren't the lines "QP" and "RS" going to be parallel, irrespective of the angle?
I am not sure why should we prove that angle C is equal to 90?[/quote]

Let me ask you: say angle c is 30 degrees, would the lines be parallel in this case?[/quote]

Thank you for the quick help Bunuel
No, they will not be parallel; but is 30 deg even possible?
Since both the points are marked "C" they should be equal. Also, since both have to be equal, 90 deg is the only option, as one will become obtuse if the other is acute and vice versa, which is not allowed (as both are marked "C").

Hence, the question answers itself, without any statement.... I think.

Please let me know where I am going wrong.

You misplaced angle c

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Yes, I realized this and edited my post. Sorry

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