enigma123 wrote:

Is the measure of one of the interior angles of quadrilateral ABCD equal to 60 degrees?

(1) Two of the interior angles of ABCD are right angles.

(2) The degree measure of angle ABC is twice the degree measure of angle BCD

Target question: Is the measure of one of the interior angles of quadrilateral ABCD equal to 60?Key concept: the 4 angles in a quadrilateral must add to 360 degrees Statement 1: Two of the interior angles of ABCD are right angles. Let's test some possible cases.

There are infinitely many quadrilaterals that satisfy statement 1. Here are two:

Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is

YES, one of the angles IS 60°Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is

NO, one of the angles is NOT 60°Since we cannot answer the

target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The degree measure of angle ABC is twice the degree measure of angle BCD.Let's test some possible cases.

Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is

YES, one of the angles IS 60°Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is

NO, one of the angles is NOT 60°Since we cannot answer the

target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined IMPORTANT: Notice that I was able to use the

same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.

In other words,

Case a: the 4 angles in ABCD are 90°, 90°, 60°, and 120°. In this case, the answer to the target question is

YES, one of the angles IS 60°Case b: the 4 angles in ABCD are 45°, 90°, 90° and 135°. In this case, the answer to the target question is

NO, one of the angles is NOT 60°Since we cannot answer the

target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,

Brent

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