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What is the degree measure of angle BAO?

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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Thu May 04, 2017 4:24 am
Image

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60º
(2) The degree measure of angle BCO is 40º

Target question: What is the degree measure of ∠BAO?

Given: The length of line segment AB is equal to the length of line sement OC

Statement 1: The degree measure of angle COD is 60º
So, we have the following:
Image

Since the radii must have equal lengths, we can see that OB = OC
Image

So, ∆ABO is an isosceles triangle.
Image

If we let ∠BAO = x degrees, then we can use the facts that ∆ABO is isosceles and that angles must add to 180º to get the following:
Image

Since angles on a LINE must add to 180º, we know that ∠OBC = 2x
Image

Now, we can use the facts that ∆BCO is isosceles and that the angles must add to 180º to get the following:
Image

Finally, we can see that the 3 angles with blue circles around them are on a line.
Image
So, they must add to 180 degrees.
We get: x + (180-4x) + 60 = 180
Simplify: 240 - 3x = 180
Solve to get: x = 20
In other words, ∠BAO = 20º
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The degree measure of angle BCO is 40º
So, we have the following:
Image

Since the radii must have equal lengths, we can see that OB = OC
Image

So, ∆BCO is an isosceles triangle, which means OBC is also 40º
Image

Since angles on a line must add to 180 degrees, ∠ABO = 140º
Image

Finally, since ∆ABO is an isosceles triangle, the other two angles must each be 20º
Image
As we can see, ∠BAO = 20º
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by GMATGuruNY » Thu May 04, 2017 4:31 am
Image

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60.
(2) The degree measure of angle BCO is 40.
It is given that AB=OC.
Since OC and OB are both radii, OC=OB.
Thus:
Image

EVALUATE THE EASIER STATEMENT FIRST.
Since statement 2 gives information about one of the equal angles, start with statement 2.

Statement 2: The degree measure of angle BCO is 40.
The result is the following combination of angles:
Image
Thus, angle BAO = 20.
SUFFICIENT.

Statement 1: The degree measure of angle COD is 60.
In the combination of angles yielded by statement 2, angle COD = 60.
Thus, statement 1 implies the same combination of angles as does statement 2.
SUFFICIENT.

The correct answer is D.
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by Turksonaaron » Mon Jun 26, 2017 11:36 pm
GMATGuruNY wrote:
Image

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60.
(2) The degree measure of angle BCO is 40.
It is given that AB=OC.
Since OC and OB are both radii, OC=OB.
Thus:
Image

EVALUATE THE EASIER STATEMENT FIRST.
Since statement 2 gives information about one of the equal angles, start with statement 2.

Statement 2: The degree measure of angle BCO is 40.
The result is the following combination of angles:
Image
Thus, angle BAO = 20.
SUFFICIENT.

Statement 1: The degree measure of angle COD is 60.
In the combination of angles yielded by statement 2, angle COD = 60.
Thus, statement 1 implies the same combination of angles as does statement 2.
SUFFICIENT.

The correct answer is D.
Hi Mitch

I get the fact that the answer is D. I was looking at your solution to get a faster route to solve it. However, I don't really understand your explanation of statement 1 which makes it sufficient.

Thanks
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by GMATGuruNY » Wed Jun 28, 2017 6:35 am
Turksonaaron wrote:Hi Mitch

I get the fact that the answer is D. I was looking at your solution to get a faster route to solve it. However, I don't really understand your explanation of statement 1 which makes it sufficient.

Thanks
Statement 2: ∠BCO = 40º
Here, the following figure is yielded:
Image
In the figure above, ∠BAO = 20º.
SUFFICIENT.

Statement 1: ∠COD = 60º
The figure yielded by Statement 2 indicates the following:
If ∠BCO = 40º, then ∠COD = 60º.

Strategy:
Test whether it's possible for ∠BCO to be less than or greater than 40º.

Case 2: ∠BCO = 30º
Here, the following figure is yielded:
Image
In this case, ∠COD is LESS THAN 60º.

Case 3: ∠BCO = 50º
Here, the following figure is yielded:
Image
In this case, ∠COD is GREATER THAN 60º.

If ∠BCO is less than or greater than 40º, then ∠COD ≠ 60º.
Implication:
To satisfy the constraint that COD = 60º, it must be true that ∠BCO = 40º, as indicated in Statement 2.
In other words, Statement 1 implies the SAME COMBINATION OF ANGLES as Statement 2.
Thus, ∠BAO = 20º.
SUFFICIENT.

The correct answer is D.
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by Turksonaaron » Wed Jun 28, 2017 9:10 am
GMATGuruNY wrote:
Turksonaaron wrote:Hi Mitch

I get the fact that the answer is D. I was looking at your solution to get a faster route to solve it. However, I don't really understand your explanation of statement 1 which makes it sufficient.

Thanks
Statement 2: ∠BCO = 40º
Here, the following figure is yielded:
Image
In the figure above, ∠BAO = 20º.
SUFFICIENT.

Statement 1: ∠COD = 60º
The figure yielded by Statement 2 indicates the following:
If ∠BCO = 40º, then ∠COD = 60º.

Strategy:
Test whether it's possible for ∠BCO to be less than or greater than 40º.

Case 2: ∠BCO = 30º
Here, the following figure is yielded:
Image
In this case, ∠COD is LESS THAN 60º.

Case 3: ∠BCO = 50º
Here, the following figure is yielded:
Image
In this case, ∠COD is GREATER THAN 60º.

If ∠BCO is less than or greater than 40º, then ∠COD ≠ 60º.
Implication:
To satisfy the constraint that COD = 60º, it must be true that ∠BCO = 40º, as indicated in Statement 2.
In other words, Statement 1 implies the SAME COMBINATION OF ANGLES as Statement 2.
Thus, ∠BAO = 20º.
SUFFICIENT.

The correct answer is D.
Alright. Thank you
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