BTGmoderatorDC wrote: ↑Mon May 23, 2022 6:17 am
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In the figure shown, point O is the center of the semicircle and points B, C, D lie on the semicircle. If the length of line segment AB is equal to the length of line segment 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:
Since the radii must have equal lengths, we can see that OB = OC
So, ∆ABO is an isosceles triangle.
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:
Since angles on a LINE must add to 180º, we know that ∠OBC = 2x
Now, we can use the facts that ∆BCO is isosceles and that the angles must add to 180º to get the following:
Finally, we can see that the 3 angles with blue circles around them are on a line.
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:
Since the radii must have equal lengths, we can see that OB = OC
So, ∆BCO is an isosceles triangle, which means OBC is also 40º
Since angles on a line must add to 180 degrees, ∠ABO = 140º
Finally, since ∆ABO is an isosceles triangle, the other two angles must each be 20º
As we can see,
∠BAO = 20º
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer = D
