GMAT Geometry: Parallelograms

Written by Kelly Granson. Posted in GMAT Study Guide

geometryMany geometry problems concern quadrilaterals and, in particular, parallelograms. Today we will continue to examine these figures and their special cases: rectangle, rhombus, and square.

Parallelogram – Quadrilateral with opposite sides parallel.

Characteristics of a parallelogram: (i.e., knowing that a quadrilateral is a parallelogram, we can affirm that)

• Opposite sides are equal.

• Opposite angles are equal.

• Diagonals bisect angles.

• Sum of the angles adjoining one side equals 180°.

Features of parallelogram: (i.e., a quadrilateral is a parallelogram, if)

• Two of its opposite sides are equal and parallel.

• Opposite sides are equal.

• Opposite angles are equal.

• Diagonals bisect by the intersection point.

Rectangle – Parallelogram with four right angles.

Characteristics of rectangle:        11.04 copy

• All properties of a parallelogram.

• Diagonals are equal.

Features of rectangle: (a parallelogram is a rectangle, if)

• One of the angles is a right angle.

• Its diagonals are equal.

Rhombus –Parallelogram with four sides equal.

Characteristics of rhombus:         12.04 copy

• All properties of a parallelogram.

• Diagonals are perpendicular.

• Diagonals bisect its angles.

Features of rhombus: (A parallelogram is a rhombus, if)

• Two of its adjoining sides are equal.

• Its diagonals are perpendicular.

• Its diagonals are bisectors of its angles.

Square –Rectangle with four sides equal.

Characteristics of the square:        13.04 copy

• All angles are right angles.

• Diagonals are equal, perpendicular, bisected by the intersection point, and bisect its angles

Features of the square: (a rectangle is a square if it has some features of the rhombus)

It is important to remember the difference between characteristics and features. A quadrilateral may have characteristics of a parallelogram (rectangle/rhombus/square) and not necessarily be a parallelogram. You must carefully analyze the available information and never assume what is not given.

For example,

• The opposite sides of a parallelogram are parallel. That's a characteristic. However, you cannot assume because two sides of a quadrilateral are parallel that it is a parallelogram; it may be a trapezoid.

• The diagonals of a rectangle are equal, but knowing that the diagonals of any quadrilateral are equal does not permit you to conclude that it is a rectangle. It may have equal diagonals and not be a parallelogram of any sort.

• The diagonals of a rhombus are perpendicular. If the diagonals of any quadrilateral are perpendicular, however, that doesn't make it a rhombus or even a parallelogram.

Think about it:

• Is every square a rectangle?

• Is every square a rhombus?

• Is every rhombus a square?

• Is every rectangle a square?

• Is every rhombus a parallelogram?

It is important to make sure that you can answer these questions and all the similar ones you can think of relating to the likenesses and distinctions among quadrilaterals.

Consider these problems:

Example 1:

Is quadrilateral ABCD a parallelogram?

(1) Two of its sides have length = 7.

(2) Two of its opposite sides have length = 9.

(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

(D) Each statement alone is sufficient to answer the question.

(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Solution:

Statement 1 tells you that two of the four sides of a quadrilateral are equal in length. But this is clearly not enough to say that this is a parallelogram. You need to know, for example, whether the two sides are opposite and parallel. Statement 2 doesn't give sufficient grounds to accept that it is a parallelogram, either, since you don't know whether these sides are parallel. Taking both statements together, however, you can conclude that with two opposite sides = 9, then the pair of length=7 must be opposite as well. Since the opposite sides of the quadrilateral are equal, then you can be sure that it is a parallelogram. The right answer is C.

Example 2:

Is quadrilateral ABCD a rectangle?

(1) Line segments AC and BD bisect one another.

(2) Angle ABC is a right angle.

(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

(D) Each statement alone is sufficient to answer the question.

(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Solution:

Statement 1 tells you that ABCD is a parallelogram but not necessarily that it is a rectangle. Statement 2 tells you that it has at least one right angle, but that is also not enough by itself to conclude that it is a rectangle.

However, a parallelogram with at least one right angle is a rectangle. The correct answer is C.

Example 3:

In the figure below, is quadrilateral PQRS a parallelogram?                    14.04 copy

(1) The area PQS is equal to the area QRS.

(2) QR = RS

(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

(D) Each statement alone is sufficient to answer the question.

(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Solution:

Statement 1 does not guarantee that the quadrilateral PQRS is a parallelogram. The common side of PQS and QRS is QS, and the equality of their areas tells you only that their heights are equal. Statement 2 also does not provide sufficient grounds to claim that PQRS is a parallelogram. Even considering the information from both statements, you can say nothing about the parallelism of the sides of the quadrilateral. The correct answer is E.

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