Geometry

[talaangoshtari]

In the figure, a, b, c, d, e, f, g, h, i, and j are chords of the circle. Which two chords are parallel to each
other?
(A) a and f
(B) b and g
(C) c and h
(D) d and i
(E) e and j

[theCEO]

Hint: the sum of the angles between the 2 points has to be 180 degrees if the lines are parallel.

[talaangoshtari]

Hint: the sum of the angles between the 2 points has to be 180 degrees if the lines are parallel.

Yes,I know that the sum should be equal to 180.My problem is in choosing those angles. For example, in determining that a and f are parallel or not, we should count which angles? ab+bc+cd+de+ef or aj+ji+ih+hg?

[theCEO]

Hint: the sum of the angles between the 2 points has to be 180 degrees if the lines are parallel.

Yes,I know that the sum should be equal to 180.My problem is in choosing those angles. For example, in determining that a and f are parallel or not, we should count which angles? ab+bc+cd+de+ef or aj+ji+ih+hg?

List the order as:
a j i h g f e d c b

going from left to right:
to check if a and f are parallel sum all angles starting at a and end at the angle before f
so to check a and f add a,j,i,h,g

to check if b and g are parallel sum all angles starting at b and end at the angle before g
so to check b and g add b,a,j,i,h

continue in similar manner for the other choices.

[theCEO]

[talaangoshtari]

Here is the answer of the book, it doesn't match...
Two lines are parallel to each other when the angle swept between them is 180°. Use this property to determine the answer:
Choice (A): Angle swept between lines a and f equals Angle between a and b + Angle between b and c + Angle between c and d + Angle between d and e + Angle between e and f = 40 + 35 + 35 + 40 + 32 = 182 ≠ 180. Hence, a and f are not parallel. Reject.
Choice (B): Angle swept between lines b and g equals Angle between b and c + Angle between c and d + Angle between d and e + Angle between e and f + Angle between f and g = 35 + 35 + 40 + 32 + 36 = 178 ≠ 180. Hence, b and g are not parallel. Reject.
Choice (C): Angle swept between lines c and h equals Angle between c and d + Angle between d and e + Angle between e and f + Angle between f and g + Angle between g and h = 35 + 40 + 32 + 36 + 45 = 188 ≠ 180. Hence, c and h are not parallel. Reject.
Choice (D): Angle swept between lines d and i equals Angle between d and e + Angle between e and f + Angle between f and g + Angle between g and h + Angle between h and i = 40 + 32 + 36 + 45 + 32 = 185 ≠ 180. Hence, d and i are not parallel. Reject.
Choice (E): Angle swept between lines e and j equals Angle between e and f + Angle between f and g + Angle between g and h + Angle between h and i + Angle between i and j = 32 + 36 + 45 + 32 + 35 = 180. Hence, e and j are parallel. Correct.
The answer is (E).

[theCEO]

Here is the answer of the book, it doesn't match...
Two lines are parallel to each other when the angle swept between them is 180°. Use this property to determine the answer:
Choice (A): Angle swept between lines a and f equals Angle between a and b + Angle between b and c + Angle between c and d + Angle between d and e + Angle between e and f = 40 + 35 + 35 + 40 + 32 = 182 ≠ 180. Hence, a and f are not parallel. Reject.
Choice (B): Angle swept between lines b and g equals Angle between b and c + Angle between c and d + Angle between d and e + Angle between e and f + Angle between f and g = 35 + 35 + 40 + 32 + 36 = 178 ≠ 180. Hence, b and g are not parallel. Reject.
Choice (C): Angle swept between lines c and h equals Angle between c and d + Angle between d and e + Angle between e and f + Angle between f and g + Angle between g and h = 35 + 40 + 32 + 36 + 45 = 188 ≠ 180. Hence, c and h are not parallel. Reject.
Choice (D): Angle swept between lines d and i equals Angle between d and e + Angle between e and f + Angle between f and g + Angle between g and h + Angle between h and i = 40 + 32 + 36 + 45 + 32 = 185 ≠ 180. Hence, d and i are not parallel. Reject.
Choice (E): Angle swept between lines e and j equals Angle between e and f + Angle between f and g + Angle between g and h + Angle between h and i + Angle between i and j = 32 + 36 + 45 + 32 + 35 = 180. Hence, e and j are parallel. Correct.
The answer is (E).

The approach that I suggested is equivalent to the answer in the book.
Lets label the point at which chord A intersects with the circle as point A.
There are two directions in which you can move 180degrees from point A and end at the same point around the circle. (you can pass through chord j or through chord b)
I went one way whereas the answer in the book chose the other way!

[talaangoshtari]

My question was actually about choosing between the direction before adding angles. From what you said I think direction is not important...