Calculating an angle in a quadrilateral - GMAT Quantitative

The four interior angles of a quadrilateral measure a total of , so we test each group of numbers to see if they have this sum.

This last group does not have the correct sum, so it is the correct choice.

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary. An isosceles trapezoid has this characteristic; this can be proved by the fact that base angles are congruent, and by the Same-Side Interior Angles Statement. For a parallelogram to have this characteristic, since opposite angles are congruent also, all angles must measure ; the rectangle fits this description, but the rhombus does not.

Case 1: The two angles are opposite angles of the parallelogram. In this case, they are congruent, and

Case 2: The two angles are consecutive angles of the parallelogram. In this case, they are supplementary, and

The diagonals of a rhombus always intersect at right angles, so . The measures of the interior angles of the rhombus are irrelevant.

Two opposite angles of a quadrilateral inscribed inside a circle are supplementary, so

The four sides of a rhombus are congruent, so a diagonal of the rhombus cuts it into two isosceles triangles. The two angles adjacent to the diagonal are congruent, so the third angle, the one marked, measures:

To prove that Polygon is also a rectangle, we need to prove that any one of its angles is a right angle.

If , then by definition of perpendicular lines, is right.

If , then, since and form a linear pair, is right.

If , then, by the Converse of the Pythagorean Theorem, is a right triangle with right angle .

If and are complementary angles, then, since

, making right.

However, since, by definition of a parallelogram, , by the Alternate Interior Angles Theorem, regardless of whether the parallelogram is a rectangle or not.

Opposite sides of a parallelogram are congruent; if , then , violating this condition.

Consecutive angles of a parallelogram are supplementary; if , then , violating this condition.

Opposite angles of a parallelogram are congruent; if , then , violating this condition.

Adjacent sides of a parallelogram, however, may or may not be congruent, so the condition that would not by itself prove that the quadrilateral is not a parallelogram.

The four interior angles of a quadrilateral measure a total of , so we test each group of numbers to see if they have this sum.

This last group does not have the correct sum, so it is the correct choice.

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary. An isosceles trapezoid has this characteristic; this can be proved by the fact that base angles are congruent, and by the Same-Side Interior Angles Statement. For a parallelogram to have this characteristic, since opposite angles are congruent also, all angles must measure ; the rectangle fits this description, but the rhombus does not.

Case 1: The two angles are opposite angles of the parallelogram. In this case, they are congruent, and

Case 2: The two angles are consecutive angles of the parallelogram. In this case, they are supplementary, and

The diagonals of a rhombus always intersect at right angles, so . The measures of the interior angles of the rhombus are irrelevant.

Two opposite angles of a quadrilateral inscribed inside a circle are supplementary, so

The four sides of a rhombus are congruent, so a diagonal of the rhombus cuts it into two isosceles triangles. The two angles adjacent to the diagonal are congruent, so the third angle, the one marked, measures:

To prove that Polygon is also a rectangle, we need to prove that any one of its angles is a right angle.

If , then by definition of perpendicular lines, is right.

If , then, since and form a linear pair, is right.

If , then, by the Converse of the Pythagorean Theorem, is a right triangle with right angle .

If and are complementary angles, then, since

, making right.

However, since, by definition of a parallelogram, , by the Alternate Interior Angles Theorem, regardless of whether the parallelogram is a rectangle or not.

Opposite sides of a parallelogram are congruent; if , then , violating this condition.

Consecutive angles of a parallelogram are supplementary; if , then , violating this condition.

Opposite angles of a parallelogram are congruent; if , then , violating this condition.

Adjacent sides of a parallelogram, however, may or may not be congruent, so the condition that would not by itself prove that the quadrilateral is not a parallelogram.

The four interior angles of a quadrilateral measure a total of , so we test each group of numbers to see if they have this sum.

This last group does not have the correct sum, so it is the correct choice.

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary. An isosceles trapezoid has this characteristic; this can be proved by the fact that base angles are congruent, and by the Same-Side Interior Angles Statement. For a parallelogram to have this characteristic, since opposite angles are congruent also, all angles must measure ; the rectangle fits this description, but the rhombus does not.

Case 1: The two angles are opposite angles of the parallelogram. In this case, they are congruent, and

Case 2: The two angles are consecutive angles of the parallelogram. In this case, they are supplementary, and