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09 May 2019, 21:10
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
39% (02:07) correct
61%
(02:25)
wrong
based on 36
sessions
History
Date
Time
Result
Not Attempted Yet
The ratio of the measures of two acute angles is 5:4, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
(A) 75
(B) 90
(C) 135
(D) 150
(E) 270
(A) 75
(B) 90
(C) 135
(D) 150
(E) 270
10 May 2019, 06:10
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This is a good question which falls in the medium-hard difficulty band of the spectrum. The best part about this question is it tests you on how well you know a very basic concept in Geometry
.
So, if ∠A + ∠B = 90, then ∠A is the complement of ∠B and ∠B is the complement of ∠A.
For example, the complement of 60 degrees is 30 degrees.
Since the two acute angles are in the ratio of 5:4, we can assume their actual values to be 5k and 4k. Then, their respective complements will be (90 – 5k) and (90 – 4k).
It is said that the complement of ONE (of 5k and 4k) is twice as large as the complement of the OTHER. Now this is where a certain degree of caution needs to be exercised because there can be two cases.
Case 1: (90-5k) is twice of (90-4k).
(90 – 5k ) = 2(90 – 4k) which on simplification yields k = 30. Then, 5k = 150 and 4k = 120. These are not acute angles. So, we will have to look at the next case now.
Case 2: (90-4k) is twice of (90-5k).
(90 – 4k) = 2(90 – 5k) which on simplification yields k = 15 (which is half of what we got in the previous case). So, quite naturally, 5k = 75 and 4k = 60, which are definitely acute angles.
Therefore, the sum of the measures of the angles is 135. So, the correct answer option is C.
In problems like these where there is a use of a phrase like ‘ONE is twice of the OTHER’, it’s a clear clue that the order is not known and multiple cases have to be taken. So, if you took up the first case and ended up with values of 150 and 120 and feel like you have hit a dead end, it’s time to take a step back and read the question again to see if you missed a case and then work out the other case.
Hope that helps!
Arvind,
CrackVerbal Prep Team
.
- Any angle whose measure is between 0 degree and 90 degree is called an acute angle.
A pair of angles which add up to 90 degrees, are called complementary angles.
So, if ∠A + ∠B = 90, then ∠A is the complement of ∠B and ∠B is the complement of ∠A.
For example, the complement of 60 degrees is 30 degrees.
Since the two acute angles are in the ratio of 5:4, we can assume their actual values to be 5k and 4k. Then, their respective complements will be (90 – 5k) and (90 – 4k).
It is said that the complement of ONE (of 5k and 4k) is twice as large as the complement of the OTHER. Now this is where a certain degree of caution needs to be exercised because there can be two cases.
Case 1: (90-5k) is twice of (90-4k).
(90 – 5k ) = 2(90 – 4k) which on simplification yields k = 30. Then, 5k = 150 and 4k = 120. These are not acute angles. So, we will have to look at the next case now.
Case 2: (90-4k) is twice of (90-5k).
(90 – 4k) = 2(90 – 5k) which on simplification yields k = 15 (which is half of what we got in the previous case). So, quite naturally, 5k = 75 and 4k = 60, which are definitely acute angles.
Therefore, the sum of the measures of the angles is 135. So, the correct answer option is C.
In problems like these where there is a use of a phrase like ‘ONE is twice of the OTHER’, it’s a clear clue that the order is not known and multiple cases have to be taken. So, if you took up the first case and ended up with values of 150 and 120 and feel like you have hit a dead end, it’s time to take a step back and read the question again to see if you missed a case and then work out the other case.
Hope that helps!
Arvind,
CrackVerbal Prep Team
06 Jun 2022, 23:23
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The sum of a complement of an angle and the angle itself is 90 degrees.
Let the angles given in the question be A and B, and their angles are 5x and 4x, respectively.
Then, the complement of angle A will be either (90 - 5x).
Similarly, the complement of angle B will be (90 - 4x).
Now as given in the question, since complement of angle B is greater than angle A(As 5x is greater than 4x),
Thus, (90 - 4x) = 2(90 - 5x)
90 - 4x = 180 - 10x
6x = 90 or x = 15.
Thus angle A will be 75, and angle B will be 60 degrees.
And the sum of angles A and B will be 135 degrees.
Let the angles given in the question be A and B, and their angles are 5x and 4x, respectively.
Then, the complement of angle A will be either (90 - 5x).
Similarly, the complement of angle B will be (90 - 4x).
Now as given in the question, since complement of angle B is greater than angle A(As 5x is greater than 4x),
Thus, (90 - 4x) = 2(90 - 5x)
90 - 4x = 180 - 10x
6x = 90 or x = 15.
Thus angle A will be 75, and angle B will be 60 degrees.
And the sum of angles A and B will be 135 degrees.
