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Which of the following must be true of the angles marked in the figure

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Which of the following must be true of the angles marked in the figure  [#permalink]

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New post 25 Dec 2018, 02:41
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A
B
C
D
E

Difficulty:

  5% (low)

Question Stats:

92% (01:06) correct 8% (01:00) wrong based on 51 sessions

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Which of the following must be true of the angles marked in the figure above?

I. a + b = d + e
II. b + e = c + f
III. a + c + e = b + d + f

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Attachment:
2018-12-25_1339.png
2018-12-25_1339.png [ 11.29 KiB | Viewed 477 times ]

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Which of the following must be true of the angles marked in the figure  [#permalink]

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New post 01 Jan 2019, 15:58
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Bunuel wrote:
Image
Which of the following must be true of the angles marked in the figure above?

I. a + b = d + e
II. b + e = c + f
III. a + c + e = b + d + f

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Attachment:
2018-12-25_1339.png


Since opposite angles are equal, let's first denote pairs of equal angles with the same symbols.
Image

Now, we'll check the statements...

I. a + b = d + e
We can rewrite this as: (red dot) + (blue star) = (blue star) + (red dot)
Definitely TRUE (always!)
Since statement I is true, we can ELIMINATE answer choice D

II. b + e = c + f
We can rewrite this as: (blue star) + (blue star) = (green square) + (green square)
This need not be true
Since statement II is false, we can ELIMINATE answer choices B and E

III. a + c + e = b + d + f
We can rewrite this as: (red dot) + (green square) + (blue star) = (blue star) + (red dot) + (green square)
Definitely TRUE (always!)
Since statement III is true, we can ELIMINATE answer choice A

Answer: C

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Which of the following must be true of the angles marked in the figure  [#permalink]

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New post 01 Jan 2019, 20:34

Solution


Given:
    • Three straight lines

To find:
    • Which of the given statements are always true?

Approach and Working:
We know that,
    • a + b + c =180, and
    • c + d + e = 180

So, a + b + c = c + d + e
    • Implies, a + b = d + e

Now, from the diagram we can say that, a + b + c = f + d + e
    • Implies, c = f

And, we can also say that, a + f + e = b + c + d
    • Since, f = c, we can interchange them.
    • Thus, we get, a + c + e = b + f + d

Hence, the correct answer is Option C

Answer: C

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Which of the following must be true of the angles marked in the figure   [#permalink] 01 Jan 2019, 20:34
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