All GMAT Math Resources
Example Question #1 : Calculating An Angle In A Quadrilateral
Note: Diagram is NOT drawn to scale.
Refer to the above diagram.
Any of the following facts alone would be enough to prove that is not a parallelogram, EXCEPT:
Any one of these facts alone would prove that is not a parallelogram.
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.
Example Question #2 : Calculating An Angle In A Quadrilateral
Which of the following can not be the measures of the four interior angles of a quadrilateral?
All four of the other choices fit the conditions.
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.
Example Question #3 : Calculating An Angle In A Quadrilateral
A circle can be circumscribed about each of the following figures except:
An isosceles trapezoid with one base three times as long as the other
An isosceles triangle with its base one-half as long as either leg
A rectangle twice as long as it is wide
A right scalene triangle
A rhombus with a angle
A rhombus with a angle
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.
Example Question #4 : Calculating An Angle In A Quadrilateral
Two angles of a parallelogram measure and . What are the possible values of ?
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
Example Question #5 : Calculating An Angle In A Quadrilateral
Rhombus has two diagonals that intersect at point ; .
What is ?
The diagonals of a rhombus always intersect at right angles, so . The measures of the interior angles of the rhombus are irrelevant.
Example Question #6 : Calculating An Angle In A Quadrilateral
Quadrilateral is inscribed in circle . . What is ?
Two opposite angles of a quadrilateral inscribed inside a circle are supplementary, so
Example Question #7 : Calculating An Angle In A Quadrilateral
Note: Figure NOT drawn to scale.
The above figure is of a rhombus and one of its diagonals. What is equal to?
Not enough information is given to answer the question.
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:
Example Question #8 : Calculating An Angle In A Quadrilateral
Refer to the above figure. You are given that Polygon is a parallelogram, but NOT that it is a rectangle.
Which of the following statements is not enough to prove that Polygon is also a rectangle?
and are complementary angles
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.