## OGQR 2020: Question No. 222

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?

Source | OGQR 2020 |

Type | Data Sufficiency |

Topic | Geometry |

Sub-Topic | Lines and Angles |

Difficulty | Easy – Medium |

### Solution

__Steps 1 & 2: Understand Question and Draw Inferences__

__Steps 1 & 2: Understand Question and Draw Inferences__

In this question, we are given:

- A portion of the roadmap
- Showing the measures of certain angles.

- All the lines shown in the figure are straight.

We need to find:

- Whether road PQ is parallel to road RS.

Per our conceptual understanding, when two parallel lines are cut by a transversal line then their corresponding angles are equal.

So, If PQ and RS are two parallel lines and line L_{1} as shown is the transversal line then the corresponding angles are equal.

- Hence, ∠1 = ∠2
- Or, c
^{0}= 180^{0 }– c^{0} - 2c = 180
- c = 90°

- Or, c

Therefore, if we get c = 90° then PQ and RS are parallel else PQ and RS are not parallel,

With this understanding, let us now analyse the individual statements.

__Step 3: Analyse Statement 1__

__Step 3: Analyse Statement 1__

*“b = 2a”*

Per conceptual understanding, the sum of all the angles of a triangle is 180.

- Hence, a + b + c = 180
- a + 2a + c = 180
- 3a + c = 180

However, from the above equation, we can get many values of a and c.

Hence, statement 1 is not sufficient to answer the question.

__Step 4: Analyse Statement 2__

__Step 4: Analyse Statement 2__

*“c = 3a”*

- a + b + c = 180
- a + b + 3a = 180
- b + 4a = 180

However, from the above equation, we can get many values of a and b.

- Hence, statement 2 is not sufficient to answer the question.

__Step 5: Combine Both Statements Together (If Needed)__

__Step 5: Combine Both Statements Together (If Needed)__

From statement 1:

- 3a + c = 180

From statement 2:

- c = 3a

Combining both the statements:

- 3a + 3a = 180.

From the above equation, we can find the value of a. Then we can find c.

Since we could find the answer by combining both the statements, option C is the correct answer.

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