Geometry - GMAT Quantitative

Card 0 of 5403

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1:

Statement 2: is an equilateral triangle.

Answer

Assume Statement 1 alone. and are a pair of vertical angles, as are and . Therefore,

By substitution,

.

Assume Statement 2 alone. The angles of an equilateral triangle all measure , so .

, , and together form a straight angle, so ,

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of ?

Statement 1: is an equilateral triangle.

Statement 2:

Answer

, , and together form a straight angle, so their measures total ; therefore,

Assume Statement 1 alone. The angles of an equilateral triangle all measure , so ; and form a pair of vertical angles, so they are congruent, and consequently, . Therefore,

But with no further information, cannot be calculated.

Assume Statement 2 alone. It follows that

Again, with no further information, cannot be calculated.

Assume both statements to be true. as a result of Statement 1, and from Statement 2, so

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. , , and together form a straight angle, so their measures total ; therefore,

However, without any further information, we cannot determine the sum of the measures of and .

Assume Statement 2 alone. , , and together form a straight angle, so their measures total ; therefore,

Again, without any further information, we cannot determine the sum of the measures of and .

Assume both statements are true. Since the measures of and can be calculated from Statements 1 and 2, respectively. We can add them:

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of ?

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. and are a pair of vertical angles and are therefore congruent, so the statement

can be rewritten as

, , and together form a straight angle, so their measures total ; therefore,

But without further information, the measure of cannot be calculated.

Assume Statement 2 alone. and are a pair of vertical angles and are therefore congruent, so the statement

can be rewritten as

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of ?

Statement 1: is a angle.

Statement 2:

Answer

Statement 1 alone gives insufficient information to find the measure of .

, , and together form a angle; therefore,

, so by substitution,

But with no further information, the measure of cannot be calculated.

Statement 2 alone gives insufficient information for a similar reason. , , and together form a angle; therefore,

Since , we can rewrite this statement as

Again, with no further information, the measure of cannot be calculated.

Assume both statements to be true. and are a pair of vertical angles, so , and . Since , then . Also,

By substitution,

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above diagram. What is the measure of ?

Statement 1:

Statement 2: is a angle.

Answer

Assume Statement 1 alone. Since and form a linear pair, their measures total . Therefore, this fact, along with Statement 1, form a system of linear equations, which can be solved as follows:

The second equation can be rewritten as

and a substitution can be made:

Assume Statement 2 alone. and are a pair of vertical angles, which have the same measure, so .

Compare your answer with the correct one above

Question

Lines_4

Note: You may assume that and are not parallel lines, but you may not assume that and are parallel lines unless it is specifically stated.

Refer to the above diagram. Is the sum of the measures of and less than, equal to, or greater than ?

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. and form a linear pair of angles, so their measures total ; the same holds for and . Therefore,

Assume Statement 2 alone. and form a linear pair of angles, so their measures total ; the same holds for and . Therefore,

, , , and are the four angles of Quadrilateral , so their degree measures total 360. Therefore,

Compare your answer with the correct one above

Question

Lines_4

Note: You may assume that and are not parallel lines, but you may not assume that and are parallel lines unless it is specifically stated.

Refer to the above diagram. Is the sum of the measures of and less than, equal to, or greater than ?

Statement 1: There exists a point such that lies on and lies on .

Statement 2: Quadrilateral is not a trapezoid.

Answer

Assume Statement 1 alone. Since exists and includes , and are one and the same—and this is . Similarly, is . This means that and have a point of intersection, which is . Since falls between and and falls between and , the lines intersect on the side of that includes points and . By Euclid's Fifth Postulate, the sum of the measures of and is less than .

Assume Statement 2 alone. Since it is given that , the other two sides, and are parallel if and only if Quadrilateral is a trapezoid, which it is not. Therefore, and are not parallel, and the sum of the degree measures of same-side interior angles and is not equal to . However, without further information, it is impossible to determine whether the sum of the measures is less than or greater than .

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Statement 1: and are complementary.

Statement 2:

Answer

Assume Statement 1 alone. and are vertical from and , respectively, so and . and form a complementary pair, so, by definition

and by substitution,

.

Assume Statement 2 alone. Since is a right triangle whose hypotenuse is times as long as a leg, it follows that is a 45-45-90 triangle, so .

, , , and together form a straight angle, so their degree measures total .

But without further information, the sum of the degree measures of only and cannot be calculated.

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. and are congruent legs of right triangle , so their acute angles, one of which is , measure . and form a pair of vertical, and consequently, congruent, angles, so .

Statement 2 alone gives insufficient information, as and has no particular relationship that would lead to an arithmetic relationship between their angle measures.

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. , , , and together form a straight angle, so their degree measures total .

Without further information, no other angle measures, including that of , can be found.

Assume Statement 2 alone. , , , and together form a straight angle, so their degree measures total .

Without further information, no other angle measures, including that of , can be found.

However, if both statements are assumed to be true, it follows from Statements 2 and 1 respectively, as seen before, that and , so

.

Compare your answer with the correct one above

Question

Lines_3

Note: Figure NOT drawn to scale.

Refer to the above figure. Give the measure of .

Statement 1:

Statement 2:

Answer

Assume both statements to be true. We show that the two statements provide insufficient information by exploring two scenarios:

Case 1:

and are vertical from and , respectively, so and , and

Case 2:

The conditions of both statements are met, but assumes a different value in each scenario.

Compare your answer with the correct one above

Question

The equations of two lines are:

Are these lines perpendicular?

Statement 1:

Statement 2:

Answer

The lines of the two equations must have slopes that are the opposites of each others reciprocals.

Write each equation in slope-intercept form:

As can be seen, knowing the value of is necessary and sufficient to answer the question. The value of is irrelevant.

The answer is that Statement 1 alone is sufficient to answer the question, but Statement 2 alone is not sufficient to answer the question.

Compare your answer with the correct one above

Question

Data Sufficiency Question

Is Line A perpendicular to the following line?

Statement 1: The slope of Line A is 3.

Statement 2: Line A passes through the point (2,3).

Answer

To determine if two lines are perpendicular, only the slope needs to be considered. The slopes of perpendicular lines are the negative reciprocals of each other. Knowing a single point on the line is not sufficient, as an infinite number of lines can pass through and individual point.

Compare your answer with the correct one above

Question

Transversal

Refer to the above figure.

True or false:

Statement 1:

Statement 2:

Answer

Statement 1 alone establishes by definition that , but does not establish any relationship between and .

By Statement 2 alone, since alternating interior angles are congruent, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. By Statement 2, . and are corresponding angles formed by a transversal across parallel lines, so . is not a right angle, so .

Compare your answer with the correct one above

Question

Untitled

Refer to the above figure. True or false:

Statement 1: is equilateral.

Statement 2: Line bisects .

Answer

Statement 1 alone establishes nothing about the angle makes with , as it is not part of the triangle. Statement 2 alone only establishes that .

Assume both statements are true. Then is an altitude of an equilateral triangle, making it - and - perpendicular with the base - and .

Compare your answer with the correct one above

Question

Untitled

Refer to the above figure. True or false:

Statement 1:

Statement 2: Line bisects .

Answer

Assume Statement 1 alone. Then, as a consequence of congruence, and are congruent. They form a linear pair of angles, so they are also supplementary. Two angles that are both congruent and supplementary must be right angles, so .

Assume Statement 2 alone. Then , but without any other information about the angles that or make with , it cannot be determined whether or not.

Compare your answer with the correct one above

Question

UntitledStatement 1:

Refer to the above figure. Are the lines perpendicular?

Statement 1:

Statement 2:

Answer

Assume Statement 1 alone. The measure of one of the angles formed is

degrees.

Assume Statement 2 alone.

By substituting for , one angle measure becomes

The marked angles are a linear pair and thus their angle measures add up to 180 degrees; therefore, we can set up an equation:

or

yields illegal angle measures - for example,

yields angle measures for both angles; the angles are right and the lines are perpendicular.

Compare your answer with the correct one above

Question

Lines

Refer to the above figure. Jane chose one of the line segments shown in the above diagram but she will not reveal which one. Which one did she choose?

Statement 1: One of the endpoints of the line segment is .

Statement 2: The line segment includes .

Answer

If we know both statements, then we know that the segment can be either or , since each has endpoint and each includes ; we can not eliminate either, however.

Compare your answer with the correct one above

Question

How many times does and intersect?

I) is a linear equation with a slope of .

II) is quadratic equation with a vertex at .

Answer

When we have a linear equation and a quadratic equation there are only so many times they can intersect. They can intersect 0 times, once, or twice.

I) Gives us the slope of one equation.

II) Gives us the vertex of our quadratic equation.

If you draw a picture, it should be apparent that we don't have enough information to know exactly how many times they intersect. Our quadratic could be facing up or down, and our linear equation could go straight through both arms, or it could miss it entirely. Therfore, neither statement is sufficient.

Compare your answer with the correct one above

Tap the card to reveal the answer