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Not sure how to add this in a hide mode, this is the OA

#

1. Since EC = 2AC, EA = CA, EC = 2(6) = 12 and line AB is an angle bisector of angle EBC. This means that angle ABC = angle ABE. Since we know that angle ABC = 30, we know that angle ABE = 30. Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90. 2. Since all the interior angles of a triangle must sum to 180: angle ABC + angle BCA + angle BAC = 180 30 + angle BCA + 90 = 180 angle BCA = 60 3. Since all the interior angles of a triangle must sum to 180: angle BCA + angle ABC + angle ABE + angle AEB = 180 60 + 30 + 30 + angle AEB = 180 angle AEB = 60 4. This means that triangle BCA is an equilateral triangle. 5. To find the area of triangle BCE, we need the base (= 12 from above) and the height, i.e., line AB. Since we know BC and AC and triangle ABC is a right triangle, we can use the Pythagorean theorem on triangle ABC to find the length of AB. 62 + (AB)2 = 122 AB2 = 144 - 36 = 108 AB = 1081/2 6. Area = .5bh Area = .5(12)(1081/2) = 6*1081/2 7. Statement (1) is SUFFICIENT

# Evaluate Statement (2) alone.

1. The sum of the interior angles of any triangle must be 180 degrees. DCG + GDC + CGD = 180 60 + 30 + CGD = 180 CGD = 90 Triangle CGD is a right triangle. 2. Using the Pythagorean theorem, DG = 1081/2 (CG)2 + (DG)2 = (CD)2 62 + (DG)2 = 122 DG = 1081/2 3. At this point, it may be tempting to use DG = 1081/2 as the height of the triangle BCE, assuming that lines AB and DG are parallel and therefore AB = 1081/2 is the height of triangle BCE. However, we must show two things before we can use AB = 1081/2 as the height of triangle BCE: (1) lines L and M are parallel and (2) AB is the height of triangle BCE (i.e., angle BAC is 90 degrees). 4. Lines L and M must be parallel since angles FDG and CGD are equal and these two angles are alternate interior angles formed by cutting two lines with a transversal. If two alternate interior angles are equal, we know that the two lines that form the angles (lines L and M) when cut by a transversal (line DG) must be parallel. 5. Since lines L and M are parallel, DG = the height of triangle BCE = 1081/2. Note that it is not essential to know whether AB is the height of triangle BCE. It is sufficient to know that the height is 1081/2. To reiterate, we know that the height is 8 since the height of BCE is parallel to line DG, which is 1081/2. 6. Since we know both the height (1081/2) and the base (CE = 12) of triangle BCE, we know that the area is: .5*12*1081/2 = 6*1081/2 7. Statement (2) alone is SUFFICIENT.

# Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

I see more assumptions than statements in the following part

and line AB is an angle bisector of angle EBC. This means that angle ABC = angle ABE. Since we know that angle ABC = 30, we know that angle ABE = 30. Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90. _________________

I see more assumptions than statements in the following part

and line AB is an angle bisector of angle EBC. This means that angle ABC = angle ABE. Since we know that angle ABC = 30, we know that angle ABE = 30. Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90.

Agreed. Point B can easily be shifted to the left and still comply with all the data, but the angle will not be 90. _________________

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