# GMAT Geometry: Parallelograms

Many 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: *

• 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: *

• 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: *

• 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?

(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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