GMAT Math : Geometry

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穰mple Questions

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穰mple Question #5 :Dsq: Calculating An Angle In A Quadrilateral

Given a quadrilateral QUAD , can a circle be circumscribed about it?

Statement 1: Quadrilateral QUAD is an isosceles trapezoid.

Statement 2: $m\angle Q = 120^{\circ }, m \angle A = 60^{\circ }$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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. Assume without loss of generality that $\angle Q, \angle U$ and $\angle A, \angle D$ are the pairs of base angles.

Then, since base angles are congruent, $m\angle Q =m \angle U$ and $m\angle A =m \angle D$ . Since the bases of a trapezoid are parallel, from the Same-Side Interior Angles Theorem, $\angle Q$ and $\angle D$ are supplementary, and, subsequently, so are $\angle Q$ and $\angle A$ , as well as $\angle U$ and $\angle D$ .

If $m\angle Q = 120^{\circ }, m \angle A = 60^{\circ }$ , then $\angle Q$ and $\angle A$ form a supplementary pair, as their measures total $180^{\circ }$ ; since the measures of the angles of a quadrilateral total $360^{\circ }$ , the measures of $\angle U$ and $\angle D$ also total $180^{\circ }$ , making them supplementary as well.

Therefore, it follows from either statement that both pairs of opposite angles are supplementary, and that a circle can be circumscribed about the quadrilateral.

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穰mple Question #6 :Dsq: Calculating An Angle In A Quadrilateral

Rectangle

The above shows Parallelogram ABCD . Is it a rectangle?

Statement 1: $m \angle 1 = 90 ^{\circ }$

Statement 2: $(AB)^{2} + (BC)^{2} = (AC)^{2}$

Possible Answers:

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

If we assume Statement 1 alone, that $m \angle 1 = 90^{\circ }$ , then, since $\angle 1$ and $\angle BCD$ form a linear pair, $\angle BCD$ is right.

If we assume Statement 2 alone, that $(AB)^{2} + (BC)^{2} = (AC)^{2}$ , it follows from the converse of the Pythagorean Theorem that $\Delta ABC$ is a right triangle with right angle $\angle B$ .

Either way, we have proved that the parallelogram is a rectangle.

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穰mple Question #7 :Dsq: Calculating An Angle In A Quadrilateral

Rectangle

Refer to the above figure. You are given that Polygon ABCD is a parallelogram butnotthat it is a rectangle. Is it a rectangle?

Statement 1: $\angle BAD \cong \angle 1$

Statement 2: $\angle DAC$ and $\angle DCA$ are complementary angles.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

It is necessary and sufficient to prove that one of the angles of the parallelogram is a right angle.

Assume Statement 1 alone - that $\angle BAD \cong \angle 1$ . $\angle BAD$ and $\angle D$ are supplementary, since they are same-side interior angles of parallel lines. Since $\angle BAD \cong \angle 1$ , $\angle 1$ is also supplementary to $\angle D$ . But as corresponding angles of parallel lines, $\angle 1 \cong \angle D$ . Two angles that are conruent and supplementary are both right angles, so $\angle D$ is a right angle.

Assume Statement 2 alone - that $\angle DAC$ and $\angle DCA$ are complementary angles, or, equivalently, $m \angle DAC + m \angle DCA = 90 ^{\circ }$ . Since the angles of a triangle have measures that add up to $180^{\circ}$ , the third angle of $\Delta DAC$ , which is $\angle D$ , measures $90 ^{\circ }$ , and is a right angle.

Either statement alone proves $\angle D$ a right angle and subsequently proves ABCD a rectangle.

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穰mple Question #8 :Dsq: Calculating An Angle In A Quadrilateral

True or false: Quadrilateral QUAD is a rectangle.

Statement 1: $\bigtriangleup QUA \cong \bigtriangleup ADQ$

Statement 2: $\bigtriangleup QUA \cong \bigtriangleup UQD$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. By congruence, $\overline{QU} \cong \overline{AD}$ and $\overline{QD} \cong \overline{AU}$ , making Quadrilateral QUAD a parallelogram. However, no clue is given to whether any angles are right or not, so whether the quadrilateral is a rectangle or not remains open.

Assume Statement 2 alone. By congruence, opposite sides $\overline{QD} \cong \overline{UA}$ , but no clue is provided as to the lengths of opposite sides $\overline{Q U}$ and $\overline{DA}$ . Also, $\angle QUA \cong \angle UQD$ , but no clue is provided as to whether the angles are right. A rectangle would have both characteristics, but so would an isosceles trapezoid with legs $\overline{Q U}$ and $\overline{DA}$ .

Assume both statements are true. Quadrilateral QUAD is a parallelogram as a consequence of Statement 1. Since $\angle QUA$ and $\angle UQD$ are consecutive angles of the parallelogram, they are supplementary, but they are also congruent as a consequence of Statement 2. Therefore, they are right angles, and a parallelogram with right angles is a rectangle.

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穰mple Question #9 :Dsq: Calculating An Angle In A Quadrilateral

True or false: Quadrilateral QUAD is a rectangle.

Statement 1: $\angle Q$ and $\angle A$ are right angles.

Statement 2: $\overline{QU} \cong \overline{AD}$

Possible Answers:

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone is insufficient to answer the question; A quadrilateral in which $\angle Q$ and $\angle A$ are right angles, $m \angle U = 91^{\circ}$ , and $m \angle D= 89^{\circ}$ fits the statement, as well as a rectangle, which by defintion has four right angles.

Statement 2 alone is insufficient as well, as a parallelogram with acute and obtuse angles, as well as a rectangle, fits the description.

Assume both statements, and construct diagonal $\overline{UD}$ to form two triangles $\bigtriangleup QUD$ and $\bigtriangleup ADU$ . By Statement 1, both triangles are right with congruent legs $\overline{QU}, \overline{AD}$ , and congruent hypotenuses, both being the same segment $\overline{UD}$ . By the Hypotenuse Leg Theorem, $\bigtriangleup QUD \cong \bigtriangleup ADU$ . By congruence, $\overline{QD} \cong \overline{AU}$ . The quadrilateral, having two sets of congruent opposite sides, is a parallelogram; a parallelogram with right angles is a rectangle.

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穰mple Question #1 :Dsq: Calculating Whether Quadrilaterals Are Similar

True or false: Rhombus $ABCD \sim$ Rhombus EFGH .

Statement 1: $\angle A \cong \angle E$

Statement 2: $\frac{AC}{EG} = \frac{BD}{FH}$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. A rhombus being a parallelogram, its opposite angles and its adjacent angles are supplementary. From this fact and Statement 1 alone, it follows that

$\angle A \cong \angle E$ , $\angle B \cong \angle F$ , $\angle C\cong \angle G$ , and $\angle D \cong \angle H$ .

By definition of a rhombus, all of its sides are congruent. By substitution,

$\frac{AB}{EF} = \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}$ .

All side proportions hold as well as all angle congruences, so the similarity statement holds.

Assume Statement 2 alone. Construct the diagonals of the rhombuses, as follows:

Rhombuses

In each rhombus, the diagonals are each other's perpendicular bisector. If

$\frac{AC}{EG} = \frac{BD}{FH}$

then

$\frac{AC \div 2}{EG \div 2} = \frac{BD \div 2}{FH \div 2}$

$\frac{AM}{EN} = \frac{BM}{FN}$

Since $\angle AMB \cong \angle ENF$ , both angles being right, it follows via the Side-Angle-Side Smiilarity Theorem that

$\bigtriangleup MAB \sim \bigtriangleup NEF$ ,

and , by similarity,

$m \angle MAB = m \angle NEF$ .

By a similar argument,

$m \angle MAD = m \angle NEH$ ,

and by angle addition,

$m \angle BAD = m \angle FEH$ .

As with Statement 1 alone, congruence of one set of corresponding angles in two rhombuses leads to the similarity of the two.

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穰mple Question #2 :Dsq: Calculating Whether Quadrilaterals Are Similar

What is the perimeter of Rhombus ABCD ?

Statement 1: AC = 6

Statement 2: BD = 8

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Each statement gives the length of one diagonal of the rhombus. Knowing one diagonal is not enough to give the perimeter of the rhombus.

Knowing the lengths of both diagonals, which is the case if both statements are assumed, is enough to determine the perimeter. The rhombus in question, along with its diagonals, is as shown below:

Rhombus

As marked in the diagram, the diagonals are perpendicular, and they are also are each other's bisector. It follows from the given lengths that AM = 3 and BM = 4 , so可以使用勾股定理计算。The perimeter is four times, since all sides of a rhombus are congruent.

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穰mple Question #3 :Dsq: Calculating Whether Quadrilaterals Are Similar

Isosceles Trapezoid TRAP has bases $\overline{TR}$ and $\overline{AP}$ .

Isosceles Trapezoid USBQ has bases $\overline{US}$ and $\overline{BQ}$ .

True or false:

Trapezoid $TRAP \sim$ Trapezoid USBQ

Statement 1: $\overline{TP} \cong \overline{UQ}$

Statement 2: $\overline{RA} \cong \overline{SB}$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

Assume both statements are true. We show that they provide insufficient information by examining two scenarios.

If Trapezoid $TRAP \cong$ Trapezoid USBQ , then $\overline{TP} \cong \overline{UQ}$ and $\overline{RA} \cong \overline{SB}$ , so the conditions of both statements are met; also, since the trapezoids are congruent, they are also similar.

Now examine isosceles trapezoid TRAP below, in which,,, andare positioned on the bases so that $\overline{TP} || \overline{UQ}$ and $\overline{RA} || \overline{SB}$ .

Trapezoid_2

Since $\overline{TP} || \overline{UQ}$ and $\overline{TU} || \overline{PQ}$ , Quadrilateral TUPQ is a parallelogram, and $\overline{TP} \cong \overline{UQ}$ ; similarly, $\overline{RA} \cong \overline{SB}$ . Therefore, Trapezoid USBQ is also isosceles, and the conditions of both statements are met. However, corresponding sides are not in proportion, since $\overline{TP} \cong \overline{UQ}$ , but $\overline{TR} \ncong \overline{US}$ ; consequently, the trapezoids are not similar.

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穰mple Question #4 :Dsq: Calculating Whether Quadrilaterals Are Similar

True or false: Rhombus $RHOM \sim$ Rhombus SIPN .

Statement 1: $OM \cdot 7 = PN$ and $HO \cdot 7 = IP$

Statement 2: The area of Rhombus SIPN is 49 times that of Rhombus RHOM .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that corresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary—that is, their angle measures total $180^{\circ }$ . Therefore, it is necessary and sufficient to prove justonepair of corresponding angles congruent.

The two statements together give no information about the measures of any of the angles of the rhombus. Therefore, together, they do not answer the question of whether they are similar or not.

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穰mple Question #5 :Dsq: Calculating Whether Quadrilaterals Are Similar

True or false: Rhombus $RHOM \sim$ Rhombus SIPN .

Statement 1: $\angle R$ and $\angle I$ are both $60^{\circ }$ angles.

Statement 2: The area of Rhombus RHOM is four times that of Rhombus SIPN .

Possible Answers:

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

声明1单独是充分的回答这个问题tion, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that coresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary - that is, their angle measures total $180^{\circ }$ . Therefore, it is neccessary and sufficient to prove justonepair of corresponding angles congruent.

Assume Statement 1 alone. $\angle R$ is a $60^{\circ }$ angle, so any angle consecutive to it, which includes $\angle H$ , is supplementary to it—that is, the angle measures total $180^{\circ }$ . This makes $\angle H$ a $120^{\circ }$ angle. Its corresponding angle in Rhombus SIPN is $\angle I$ , which is a $60^{\circ }$ angle. Since $\angle H$ and $\angle I$ are noncongruent, it follows that Rhombus $RHOM \nsim$ Rhombus SIPN . (Note that it can be demonstrated that the rhombusesaresimilar, but the correct statement is Rhombus $RHOM \nsim$ Rhombus IPNS .)

Statement 2 alone provides no useful information; the relationship between the areas of the rhombuses is irrelevant.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

穰mple Questions

穰mple Question #5 :Dsq: Calculating An Angle In A Quadrilateral

穰mple Question #6 :Dsq: Calculating An Angle In A Quadrilateral

穰mple Question #7 :Dsq: Calculating An Angle In A Quadrilateral

穰mple Question #8 :Dsq: Calculating An Angle In A Quadrilateral

穰mple Question #9 :Dsq: Calculating An Angle In A Quadrilateral

穰mple Question #1 :Dsq: Calculating Whether Quadrilaterals Are Similar

穰mple Question #2 :Dsq: Calculating Whether Quadrilaterals Are Similar

穰mple Question #3 :Dsq: Calculating Whether Quadrilaterals Are Similar

穰mple Question #4 :Dsq: Calculating Whether Quadrilaterals Are Similar

穰mple Question #5 :Dsq: Calculating Whether Quadrilaterals Are Similar

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