Lines and Angles

An angle is a measure of rotation. It is composed of two rays that meet in a single point. An angle is marked by the "∠" sign. It is measured in degrees, marked "°".

An angle may be denoted by a lower case letter or by three points that determine it.

A closed plane figure for which all sides are line segments is a polygon.

The name means numerous sides. A polygon which has all sides mutually congruent and all angles mutually congruent is called a regular polygon or equilateral polygon. Most polygons in GMAT Geometry problems will be regular.

A diagonal is a line that joins two corners which aren't already connected by a side.

The number of diagonals in a polygon (a figure with numerous sides) is given in the formula:n(n-3) / 2, where n is the number of sides.

Congruent means exactly equal in size and shape.

Congruent sides or segments have the exact same length. Congruent angles have the exact same measure.

ABCDEF is a regular six sided polygon, and EFG is a regular three sided polygon. If point G is not found inside ABCDEF, then ∠AFG is a straight angle an acute angle an obtuse angle a right angle a 360° angle

A Correct! Draw the hexagon and place point G outside of it. To create an equilateral triangle with point G at its vertex, you must draw straight lines from points E and F. In other words, ∠AFG must be a straight angle.

Q 6...The number of vertices formed by four lines could be any number from 0 to 6 1 to 6 4 to 6 0 to 4 4 to 5

A Correct! Since there is no quick or formulaic way to solve this question, use POE. Since 4 parallel lines create no vertices, eliminate all answer choices that do not include 0, leaving you with A and D. Next, eliminate answer choice D because it is possible to draw a figure with 6 vertices out of 4 lines, as shown below.

Q10...If in the figure above CMD is 110° and CM is a bisector of AMB, then what is x? 40° 55° 70° 140º The information is insufficient to determine x.

A Correct! ∠AMD is a straight angle and therefore all its component angles are supplementary. CM is a bisector of ∠AMB and therefore cuts the angle in half. Therefore, ∠AMC and ∠CMB are equal. You know that ∠CMD is 110º, and therefore its supplementary angle, ∠AMC, is 70º. This means that ∠CMB is also 70º and ∠AMB is 140º. Since ∠AMB+x=180º, x=40º.

Lines and angles are the building blocks of GMAT Geometry. Almost every Geometry problem in the test is comprised of lines and angles.

A line has no width and extends infinitely in either side. A line segment is determined by 2 points, and denoted by capital letters or a single lower case letter. (i.e. line AB, line m) If a line or a line segment seems straight, you may assume it is straight. If a line or a line segment passes through a point, you may assume it does.

(√7−√5)²=? 2−2√35 12−2√35 √2 2 24

B Correct. Use recycled quadratic II: (a-b)2 = a2-2ab+b2 --> (√7−√5)2 = --> (√7)2−2×√7×√5+(√5)2 = --> 7-2√7×√5+5 = --> 12-2√35

Length is the distance from one point to another. The term length can also apply to a dimension of a certain two or three dimensional figure (e.g. length and width are the two dimensions of a rectangle.)

The length is measured in feet, inches, centimetres etc.

Two angles are supplementary if they add up to 180°, or to a straight angle.

However, the angles don't have to be adjacent, as long as they add up to 180°.

Two angles are complementary if they add up to 90°, or to a right angle.

However, the angles don't have to be adjacent, as long as they add up to 90°.

Q19...If a>b in the figure below, (the figure is not drawn to scale,) then which of the following must be true? x<0 x=0 x>0 There is no x that fits the information in the question. The information in the question is not sufficient to determine which of the answers A-C is true.

A Correct. Remember that in any triangle the shortest side is opposite the smallest angle and, likewise, the longest side is opposite the largest angle. Given that a>b, the angle opposite a is more than the angle opposite b i.e. (40 + 2x) > (40 + 3x). Based on this, 2x > 3x or 0 > 3x - 2x. Hence, 0 > x.

Q 11.....In the figure below, what is the value of x? 10° 15° 20° 25° 30°

A Correct. Remember that the sum of the angles inside any triangle is 180°. Also, angles on a straight line sum up to 180°. Since sum of all angles in a triangle is 180°, 90° + 50° + 30° + x = 180° i.e. x = 180° - 90° - 50° - 30° = 10°.

What is the largest number of infinite lines that intersect at exactly two vertices? 3 4 5 6 Any number of lines may intersect at exactly two vertices.

A Good! A vertex is the intersection point of at least two straight lines. Begin with 2 lines intersecting at one vertex, then try to add lines, while keeping to a maximum of two vertices. Since the problem doesn't supply a figure, draw one yourself. Three lines may intersect at two vertices. Adding another line inevitably results in at least one more intersection, Even if the additional line passes through an existing intersection, as shown below, 3 is the maximum number of lines that can intersect at just two vertices.

51/2413 is approximately: 0.2% 0.25% 0.5% 1.5% 2%

A Incorrect. Percent problems are always a wonderful opportunity to ballpark. Avoid tedious calculations by replacing uncomfortable "ugly" numbers with rough estimates. B Incorrect. E Correct. 51/2413 is approximately 50/2500 or 1/50, which is 2/100 or 2%.

In a certain aquarium, the number of red fish is three times the number of the green fish, and the number of blue fish is half the number of the red fish. If the ratio of green fish to red fish were to be doubled, and the ratio of red fish to blue fish were to be doubled, then the ratio of blue fish to green fish in that aquarium would be 3:2 1:1 1:2 3:4 3:8

A Incorrect. There are two issues in this question. At first, you must form a new ratio by multiplication/division of another ratio. Then, you should combine the original ratio and the new formed ratio. Break the problem into two steps, while organizing the information in the ratio box. First, write down the initial ratios. Go on to double the ratio of green to red, and pay attention to the fact that the green is mentioned first. Double the red to blue ratio as well. Next, re-combine the ratios by equating the number that represents the common quantity, the red fish. Expand or reduce ratios if needed. E Correct. Here are the two ratios Red : Green : Blue 1st ratio 3 : 1 : 3/2 Expand the ratio by 2 to work with easier numbers: Red : Green : Blue 1st ratio 6 : 2 : 3 Double the green:red ratio means that there should be twice as many green fish for the same previous number of red fish - basically, double the green portion only to get 6 : 4. By the same rationale, Double the red:blue ratio means doubling only the red side to get 12 : 3. Next, combine the two ratios. Red : Green : Blue 1st ratio ( 6 : 4 ) /·2 2nd ratio ( 12 : 3 ) Combine the two ratios by equating the common member - red. Expand the 1st ratio by 2 to get Red : Green : Blue 1st ratio ( 12 : 8 ) 2nd ratio ( 12 : 3 ) Thus, the ratio of blue to green fish is 3:8.

A side is the straight line connecting two adjacent vertices (corners) of a polygon (a figure with three sides or more). The sum of the lengths of the sides of any polygon is the perimeter, the boundary line.

A perimeter is always a closed line encircling the full figure. Think of a wall surrounding a castle in the shape of the figure - the perimeter is the length of that wall - the distance you would cover until you get back to the point where you started.

An angle is acute if it measures less than 90°.You can remember this term by thinking of a cute angle - a small, less than 90°, slightly furry angle. (Thanks, Buffy!) The term acute is also used to describe someone that is quick to understand.

A right angle measures exactly 90°. If you see in a GMAT problem two lines marked by a "box" (shown in the figure), you may assume the lines form a 90° angle. Two lines that intersect at a right angle are perpendicular to each other. It is a relative term, marked by "|".

An angle is obtuse if it is greater than 90° and smaller than 180°. The term obtuse is also used to describe someone who is slow to understand.

Angle AMB is a right angle, angle AMC is obtuse and angle BMC is acute. What is ∠AMC−∠BMC ? 45° 90° 135° 120° 180° B Correct. Draw your own figure when the question does not give a figure. Below is the only configuration of A, B, M, and C which fits the question. Angle AMC = 90° + ∠BMC so ∠AMC - ∠BMC = 90°.

The perimeter of a polygon with sides of integer length is 45. If the smallest side of the polygon is 5 and the longest side of the polygon is 10, then the number of sides could be any number from 5 to 7 5 to 8 5 to 9 6 to 8 6 to 9

B Correct. Because two sides must measure 5 and 10, subtract 5+10=15 from the perimeter 45 to get 30. The minimum number of sides will occur if all the sides (except the shortest one) will be the same size as the longest one, i.e all sides are 10. 30/10=3. Adding the two sides of 5 and 10, the minimum number of sides is 5: {5, 10, 10, 10, 10}. The maximum number of sides will occur if all the sides (except the longest one) will be the same size as the shortest one, i.e. all sides are 5. 30/5=6. Adding the two sides of 5 and 10, the maximum number of sides is 8: {5, 5, 5, 5, 5, 5, 5, 10}.

Q2...If in the figure above, l1 and l2 are parallel, what is x? 30° 40° 50° 55° 70°

B Correct. Lines given in a parallel lines diagram can be extended to make it easier to solve the question. In this case draw a line parallel to l1 and l2 in the middle to divide the 110 angle in the middle of the figure. Draw a line parallel to l1 and l2 between the two parallel lines. The figure shows line l3 parallel to l1 and l2. The alternate angle of the 110 angle on l2 is also 110. Based on this, the supplementary angle of the 110 angle created as a result of l3 is 70 (180 - 110 = 70). Since the 110 angle divided in the middle is divided into 70 and 40, the alternate angle of 40 is also 40. Hence x = 40.

What is the closest approximation for 9.6% of 42? 3 4 5 30 40

B Correct. Percent problems are always a wonderful opportunity to ballpark. Avoid tedious calculations by replacing uncomfortable "ugly" numbers with rough estimates. 9.6% of 42 is roughly equivalent to 10% of 40 which is 4.

In right triangle DBC shown below, what is the value of x? 20° 30° 40° 50° 60°

B Correct. Remember that the sum of the angles inside any triangle is 180°. Also, angles on a straight line sum up to 180°. Better-look at the big triangle and add up all angles to get 180: 40+20+x+90=180. Hence, x=30.

Answer the following Non-GMAT question: If c≠−1, which of the following does NOT always equal a/b ? a×(c+1)/b×(c+1) a×a/b×b 19a/b×19 (a/55) / (b/55) 0.1a/(b/10)

B Correct. Try to expand/reduce the fractions to form a/b. If you succeed, then the fraction is equal to a/b and you can POE it. If, however, you realize that the fraction has been attained from a/b by multiplying the top and bottom by two different numbers, this must have changed the value of the fraction from a/b, and this is the correct answer. Unless a=b, multiplying the top of a fraction by a and the bottom by b, changes its value. Therefore a×a/b×b is not equal to a/b.

Which of the following describes all the values of y for which (y+5)2 < 16? −1 < y < 9 −9 < y < −1 −1 < y < 1 −3 < y < 3 −√11 < y < √11

B Correct. When x2 < a, then -√a < x < √a. When x2 > a, then x > √a or x < -√a. In this question you are dealing with the first scenario: If (y+5)2 < 16, then -√16 < y+5 < √16 -4 < y+5 < 4 --> -4 < y+5 AND y+5 < 4 Solve the two inequalities separately: Subtract 5 to isolate y. --> y > -9 AND y < -1 Combine: --> -9 < y < -1

Steven and Brandon started their companies with an equal amount of capital and invested what they earned into the companies. Steven had a deficit of 33.33% in the first year and a profit of 25% in the second year. If Brandon earned a profit of 33.33% in the first year and deficit of 25% in the second year, after two years in business Steven's assets are what percent of Brandon's assets? 66.67% 83.33% 100% 120% 133.3%

B Correct. Whenever there's an invisible variable in the problem plug in a good number. If the problem asks about percents use 100 or multiples of it. In this case, plug in a good number for Steven or Brandon's investment. Remember that percent problems are a great opportunity to ballpark. Plug in 300 for Steven's and Brandon's investments (the invisible variable). In this case 300 works better than 100, with 33.33%. If Steven invests $300, a loss of 33.33%, would leave him with 300-100=$200 after the first year, and after a profit of 25% in the second year, Steven will have 200+50=$250. If Brandon invests $300, a profit of 33.33%, would leave him with 300+100=$400 after the first year, and after a loss of 25% in the second year, Brandon will have 400-100=$300. Now translate the question: Steven's assets are what percent of Brandon's assets? 250=x/100 × 300; x≈80+% Thus, Steven's assets are 83.33% of Brandon's assets.

Q13...A shape is formed by attaching a regular hexagon to a regular pentagon, as shown above. If the perimeter of this shape is 99, what is the perimeter of the pentagon? 66 55 54 45 11

B Good work! In a regular polygon, the sides are of equal length. Since these two shapes share a border, the resulting shape also has 9 equal sides. Therefore, divide 99 by 9 to get a side length of 11. Since the pentagon has 5 sides, its perimeter is 55.

Which of the following is true? All regular five sided polygons are congruent. All polygons of perimeter 3 are congruent. All regular six sided polygons of perimeter 10 are congruent. All squares are congruent.

B Incorrect. Congruent means exactly equal in size and shape. Congruent geometric figures are identical. Perimeter is the sum of the lengths of each side of a polygon. This statement does not limit the number of sides for the polygons being compared: a square and a triangle, both with a perimeter of 3, will not be congruent. C Correct. A regular polygon must have equal sides and angles. Perimeter is the sum of the lengths of each side. Therefore if each regular six sided polygon has the same perimeter, each side must be of the same length, and the resulting regular hexagons must be identical, or congruent.

Q14....Which of the following is not true? All the diagonals of a regular four sided polygon are congruent. All the diagonals of a regular five sided polygon are congruent. All the diagonals of a regular six sided polygon are congruent.

B Incorrect. Draw a regular five sided polygon. Its diagonals are all of the same length. C Correct. In a regular hexagon, shown here, there are 3 possible diagonals from each vertex. Segment AD goes straight down the shape while segments AE and AC are at an angle to the side. You can see that AD is longer than AE or AC.

If m<0, which of the following must be true? -100 < m < 100 m ≤ -1 m ≤ 1 m2 ≥ 1/4 -100 < m < 0

B Incorrect. Plug in numbers that fit m<0. Try to disqualify the answer choices. How about m=-1/2 ? E Incorrect. Plug in numbers that fit m<0. Try to disqualify the answer choices. How about m=-200 ? D Incorrect. Plug in numbers that fit m<0. Try to disqualify the answer choices. How about m=-1/10 ? C Correct. The best way to approach variables in the answer choices, especially with "must be" questions, is by Plugging In. In "must be" questions use this rule of thumb: "If it fails - POE. If it doesn't, make it fail!" Your mission in "must be" questions is to try and disqualify the answer choices, then POE, POE, POE, until only one remains. Plug in numbers that fit m<0, then POE; Plug in m=-200 to POE A and E. Plug in m=-1/2 to POE B. Plug in m=-1/10 to POE D. The answer is C.

A, B, C and D are points on a line. If AB is 6, BC is 2 and CD is 1, then AD could not be 3 4 5 7 9

B Very good! Since there is no quick way to solve this problem, we must apply POE to each answer choice. Answer choice A (line AD=3) can be eliminated by arranging the line into the form A-D-C-B (AB-CB-DC=6-2-1). Answer choice C (line AD=5) can be eliminated by arranging the line into the form A-C-D-B (AB-CB+CD=6-2+1). Answer choice D (line AD=7) can be eliminated by arranging the line into the form A-B-D-C (AB+BC-DC=6+2-1). Answer choice E (line AD=9) can be eliminated by arranging the line into the form A-B-C-D (AB+BC+CD =6+2+1). This leaves answer choice B as the correct answer.

Q18...Two congruent triangles were combined in three different ways, as shown above. If the perimeters of the shapes above are 12, 14, and 16, then what is the perimeter of one triangle? 13/2 14/2 21/2 22/2 42/3

C Correct. Remember that congruent triangles have same angles and same sides. The perimeter of any triangle is the sum of its sides. Given that the two triangles are congruent we can name their sides as a, b and c. It doesn't really matter how you choose to label the sides, but let's go with a=height (short base), b=long base, c=hypotenuse. Based on this the perimeter of the shapes resulting from combining the triangles in different ways (as shown in the figure) is Rectangle (two long bases, two heights): 2a + 2b = 12, Parallelogram on top right of the figure (two short heights, two hypotenuses): 2a + 2c = 14 and Bottom parallelogram (two long bases, two hypotenuses): 2b + 2c = 16. We could use these three equations to find the individual values of a,b and c, but we don't really need to - we're looking for the perimeter of the triangle, which is the expression a+b+c. To find the value of that expression, add all three equations to get 4a + 4b + 4c = 42 --> 4(a + b + c) = 42 --> a + b + c = 42/4 = 21/2. Hence, the perimeter of one triangle is 21/2.

Q5... Note: Figure not drawn to scale. BM is the bisector of ∠AMC, and CM is the bisector of ∠BMD. If ∠AMB=42°, what is ∠CMD? 14° 21° 42° 63° 84°

C Correct. A bisector is a line that cuts the angle in half. BM is the bisector of ∠AMC, therefore ∠AMB equal to ∠BMC. CM is the bisector of ∠BMD, therefore ∠BMC equal to ∠CMD. Since ∠AMB=42°, ∠BMC and ∠CMD also equal 42°.

Q3...If in the figure above, l1 and l2 are parallel, what is x? 80° 90° 100° 110° 120°

C Correct. Lines given in a parallel lines diagram can be extended to make it easier to solve the question. Extend l2 backwards and extend the line intersecting both parallel lines. The resulting triangle is shown in the figure. The supplementary angle of 120 is 60. Given that l1 and l2 are parallel lines, the alternate angle of 40° is also 40°. Since the sum of all angles in a triangle is 180, the supplementary angle of x is 80=180 - 60 - 40. Hence, x = 180 - 80 = 100°.

If the complement of a certain angle is three times the measure of that certain angle, then what is the measure of that certain angle? 45° 30° 22.5° 18° 15°

C Correct. Remember the complement of an angle and the angle add up to 90°. Begin with assuming the angle to be x. Thus the complement of x is (90 - x). Translate English to Math. If the angle is x, the complement of x is (90 - x). According to the question (90 - x) = 3x or 90 = 3x + x or 90 = 4x or 90/4 = x. Hence, the angle is 22.5°. Alternatively, a specific question and numbers in the answer choices warrant Reverse PI. Start with the middle answer 22.5. Its complement is 67.5, which is exactly three times the original angle. The answer is C.

5^8− 4 × 5^7 + 4×5^6=? −11·5^6 5^7 9·5^6 81·5^6 81·5^8

C Correct. When adding or subtracting powers: DON'T: add or subtract the exponents. DO: extract the highest common factor. 5^8−4·5^7+4·5^6 = --> 5^6(5^2-(4·5^1)+4)= --> 5^6·(25-20+4)= --> 9·5^6

Q16...Polygon ABCDE is divided into three congruent triangles, as shown above. If ∠BAE=60°, what is ∠BCD? 120° 130° 140° 150° 160°

C Incorrect. The angles inside any triangle always sum up to 180° and congruent triangles have equal angles and lengths. E Yes! ∠BAE equals 60°. Since the triangles are congruent, its 3 component angles each equal 20°. Therefore, in triangle ABC, the sum of ∠ABC and ∠ACB is equal to 160°. Because triangles ABC and ACD are congruent, ∠ACD is equal to ∠ABC. You already know that the sum of ∠ABC and ∠ACB equals 160°. Therefore the sum of ∠ACD and ∠ACB also equals 160°.

Mona and Donald fly to Rome for the weekend. They take cash only in notes of $10 and notes of €10. Mona carries three times the amount of euros Donald carries. She also carries as many dollars as Donald carries. The number of €10 notes they take is double the number of $10 notes they take. If Donald carries a total of 40 notes (of either $10 or €10,) then what is the total number of notes (of either $10 or €10,) they take? 70 80 100 120 150

C Incorrect. This is a Sets question. It asks about the total number of notes Mona and Donald carry. The set of all €10 notes and the set of all $10 notes are two exclusive sets that make up the total of all cash notes they carry. The notes that Donald carries and the notes that Mona carries are also two exclusive sets that make up the total of all cash notes they carry. Draw a 3x3 "A/B Table". Use variables to represent ratios between boxes in the table. D Correct. Set the "Mona/Donald" - "$/€" table so - Mona carries three times the amount of euros Donald carries. Put 3x in the Mona / € box and x in the Donald / € box. Note that this means that the Total / € is 3x+x=4x. She also carries as many dollars as Donald carries. Put y in the Mona / $ box, and y in the Donald / $ box. Note that this means that the Total / $ box is y+y=2y. The total number of €10 notes Mona and Donald take together is double the total number of $10 notes they take together. This means that 4x = 2·2y If Donald carries a total of 40 notes (of either $10 or €10)... Put 40 in the Donald / Total box. ...then what is the total number of notes (of either $10 or €10,) they take? - Place the ? in the Total/Total box. ..........Mona....Donald......Total $10 ........y..........y..............2y €10 .......3x........x...............4x Total ................40...............? 4x = 2·2y, therefore x=y. In terms of x the table looks so ..........Mona....Donald......Total $10 ..y=x........y=x...........2x €10 .......3x........x...............4x Total ..............40=2x.....?=6x So 2x=40, and the Total / Total is ?=6x=120.

(3+√3)2=? 12+3√3 12 6+6√3 18 12+6√3

Correct. Use recycled quad I. --> (3+√3)2= --> 32+2×3×√3+(√3)2 = --> 9+6√3+3 = --> 12+6√3

Molly's bank pays 10% compound interest on savings annually. Molly deposited 10,000 dollars in her account at the start of last year, another 10,000 dollars at the start of this year and plans to deposit 10,000 dollars more at the start of next year. If she makes no other deposits to or withdrawals from the account, what will be her balance at the end of next year? 28075 29500 33100 36410 39720

D Correct. Compound interest is calculated on the principal amount, as well as on any interest already earned. For this question, calculate simple interest for each year while updating the principal amount for every year by adding principal amount and interest from last year to new deposits made by Molly. Last year Molly deposited $10,000 so she earned Interest1 = Principal x Interest rate x Time Interest1 = 10,000 x 10/100 x 1 Interest1 = 1000 This year her opening balance was 10,000 + 1,000 = 11,000 and she deposited 10,000. So interest earned this year will be applicable to 10,000 + 1000 + 10,000 = 21,000. Interest2 = 21,000 x 10/100 x 1 Interest2 = 2100 Next year her opening balance will be 21,000 + 2100 = 23,100 and she will deposit 10,000. So interest earned this year will be applicable to 21,000 + 2100 + 10,000 = 33,100. Interest3 = 33,100 x 10/100 x 1 Interest3 = 3310 Hence, her balance at the end of next year will be 33,100 + 3310 = 36410.

Which of the following describes all the values of x for which (3x+6) / (−2) > x−2? x > −10 x < −10 x > −0.4 x < −0.4 x < −2

D Correct. Manipulate the inequality until you reach one of the answer choices. Remember: when multiplying or dividing by negative number, flip the sign. (3x+6) / (-2) > x-2 Multiply by (-2). Don't forget to flip the sign: 3x+6 < (x−2)(-2) 3x + 6 < -2x + 4 Now isolate x, as you would in an equation: 5x < -2 x < -2/5 --> x < -0.4

If z≠2, and (2−z)(2z−1)=0, then z= −2 −1 0 1/2 1

D Correct. The issue of this question is understanding the meaning of the factored quadratic equation. If (2−z)(2z−1)=0, then at least one of the factors must be zero. It follows that either - (2−z)=0 --> z=2 (2z−1)=0 --> z=1/2 Which of these is the correct value for z? If z≠2, then (2−z)≠0 and (2z−1) must be zero, i.e., z=1/2.

If (√a)+(√b)=5√2, and a·b=144, what is the value of a+b? √26 √38 √338 26 38

D Correct. The question presents the sum of square roots of a and b, and the product of ab, which is itself a perfect square. The various elements presented by the question stem should remind you of Recycled quadratic I: (a+b)2 = a2+2ab+b2. Get rid of the annoying sum of squares by squaring both sides of the equation (√a)+(√b)=5√2: [(√a)+(√b)]2=(5√2)2 Expand the squared equation: (√a)2 + 2√a√b + (√b)2=52(√2)2 Squares and square roots cancel each other out, so: a + 2√a√b + b=25·2 --> a + b + 2√ab = 50 ab=144, according to the question stem, so --> a + b + 2√144 = 50 --> a + b + 2·12 = 50 Isolate a + b: --> a + b = 50 - 24 = 26

Due to a manufacturing problem, all the USB flash drives produced in a certain factory had either a hardware defect or a software defect or both. A batch of 350 USB flash drives was tested. 50 USB flash drives suffered only from the software defect, and 100 suffered from both defects. How many of the USB flash drives tested did not suffer from the software defect? 50 100 150 200 250

D Correct. This is a sets question. Since the question presents two sets with an interaction of A, B and both A and B, draw an "A/not A" 3×3 table. Place the information in the question in the table: 350 Drives were tested - that's the sample space, so it goes in the Total / Total box. 50 drives suffer only from software - means not Hardware problems, so this goes in Software / No Hardware box. 100 suffered from both defects - goes in the Software / Hardware box. Finally, what did the question ask? How many of the USB flash drives tested did not suffer from the software defect. Doesn't specify whether or not they also suffered from the hardware defect, so this goes in the Software / Total box. Put a little question mark there. The table looks as following: ..............Soft....No Soft......Total Hard ......100.......................... No Hard ..50........................... Total ....................?.............350 Now, fill in the necessary boxes: Left column: Total Software = 100 + 50 = 150. Bottom row: Total No Software = 350 - 150 = 200 Middle column: ..............Soft....No Soft......Total Hard ......100.......................... No Hard ..50........................... Total ......150........200.......350

Which of the following CANNOT be true (for any real x)? I) x2−x+2=0 II) x2−2x+1=0 III) 2x2−x+1=0 I only II only III only I and III only I, II and III

D Correct. When can't a quadratic equation be true for any real x? when it has no real solution. This happens when the discriminant (b2-4ac) is negative. According to the 1st equation, --> x2−x+2=0 --> b2-4ac=(-1)2-4·1·2=1-8=-7 The discriminant is negative, so there are no real solutions to this equation. Therefore, this equation CANNOT be true for any real x. According to the 2nd equation, --> x2−2x+1=0 --> b2-4ac=(-2)2-4·1·1=4-4=0 The discriminant=0, so there is one solution to this equation. Therefore, this equation can be true. --> b2-4ac=(-1)2-4·2·1=1-8=-7 The discriminant is negative, so there are no real solutions to this equation. Therefore, this equation CANNOT be true for any real x.

If Mark saves 50% more than Sally every month, Sally's annual savings are approximately what percent of Mark's annual savings? 25% 33% 50% 67% 75%

D Correct. Whenever there's an invisible variable in the problem plug in a good number. If the problem asks about percents use 100 or multiples of it. Remember that percents problems are a great opportunity to ballpark. It is always better to start with the smallest variable, so make Sally's monthly savings the invisible variable. When the problem asks about percents use 100. And so, If Sally saves $100 per month, Mark saves 50% more, i.e., $150. Now pause for a moment. Did you multiply Sally's and Mark's savings by 12 (months)? 1. Of course I did. How else can I answer the question? 2. No I didn't. There's no need. Ans: 2. There's really no need to multiply Mark's or Sally's savings. If Mark saves 50% more than sally does, every month, it is the same for a whole year savings. Save yourself the trouble and work with smaller numbers. The question is actually a percent translation sentence. And so, "Sally's annual savings are approximately what percent of Mark's annual savings?" translates into 100 = x/100 × 150. Reduce 150/100 by 50 to 3/2 and solve: 100 = 3x/2 /*2 200 = 3x /:3 200/3 = x to get x=200/3≈66.7

Q7...How many vertices does a polygon have, if each vertex is the intersection of exactly three diagonals? 3 4 5 6 No such polygon exists.

D Good! A diagonal is a line that joins two corners which aren't already connected by a side. In a polygon, each vertex is adjacent to two other vertices. If each vertex is the originating point for three more diagonals, the polygon has three additional vertices. The total sum of the vertices is 1 (the original vertex) + 2 (adjacent vertices) + 3 (non-adjacent vertices). For example, focus on the vertex A. The three diagonals intersecting at vertex A will lead to three vertices D, E and F - that's four vertices mandated by the requirement set in the question. However, beyond these four vertices, vertex A is also connected to adjacent vertices B and C with two sides of the polygon, leading to a total number of vertices of 6.

Q9...In the figure above, MD is the bisector of ∠AMB, and ME is the bisector of ∠AMC. What is x? 5° 7.5° 10° 15° 20°

D Incorrect. A bisector is a line that cuts an angle in half, and supplementary angles add up to 180°. C Correct. Line AM creates an angle of 180°. Therefore, ∠AMB is 150° (180°-30). Since MD is a bisector of ∠AMB, ∠AMD is a 150°/2=75°. Likewise, ∠AMC is a 130° (180°-(30°+20°)). Since ME is a bisector of ∠AMC, ∠AME is 130°/2=65°. x equals ∠AMD - ∠AME, or 75°-65°=10°.

If −y ≥ x, and −x < −5, then which of the following must be true? y = −5 y > −5 −y > 5 y ≤ −5 y ≥ −5

D Incorrect. Manipulate the inequalities until you reach one of the answer choices. Remember: when multiplying or dividing by negative number, flip the sign. C Correct. Let's get some order in the court, shall we? Focus on −x < −5, since that at least contains a real number. Multiply by -1, and don't forget to flip the sign: x > 5 According to the first inequality, -y ≥ x > 5. Therefore, -y > 5.

The perimeter of a polygon is 16. If the sides of the polygon are all of integer length, the shortest side of the polygon is 2 and the longest side of the polygon is 5, then the number of sides of the polygon could be any number from 3 to 6 4 to 5 3 to 7 4 to 6 4 to 7

D You grossly underestimated the time this question took you. You actually solved it in 5 minutes and 9 seconds. Correct. To find the maximum number of sides repeat the shortest length as many times as possible. Likewise, repeat the longest side to find the minimum number of possible sides. Remember that the shortest side must be 2 and the longest side must still be 5. To reflect the presence of these two sides, subtract the length of the two known side from the perimeter of the polygon: 16-2-5 = 9. This difference of 9 is then divided between the the lengths of the other sides. To create the maximum number of sides, choose the shortest lengths possible: the other sides can be 2, 2, 2, 3, creating a polygon with 6 sides: 2, 2, 2, 2, 3 and 5. To create the minimum number of sides, choose the longest lengths possible: the other sides can be 4, 5, creating a polygon with 4 sides: 2, 4 ,5, 5, Hence, the number of sides can range between 4 and 6.

(2^5·5^6)/10000=? 0.2 5 25/2 25 50 Alternative method: Split the 5^6 into 5·5^5 and combine like terms under the same power: (2^5·5^6)/10000 = --> 2^5·5^5·5/10000 = --> (2·5)^5·5/10000 = --> 10^5·5/10^4 = --> 5·10^1 = 50

E Correct. Convert 10000 to the form Ab to bring everything to same the base of 2 and 5. (2^5·5^6)/ 10000 = --> (2^5·5^6)/ 10^4 = --> (2^5·5^6)/ 2^4·5^4 = --> (2^5-4·5^6-4) = --> (2·5^2) = --> (2·25) = --> 50 Hence, this is the correct answer.

The following is a basic algebra practice exercise, and not a GMAT level problem: First extract the greatest common factor out of the expression below 18xz-3xy+9x= Click "continue" when you're done. Now that you have the factored form of the expression, which of the following expressions is equivalent to the expression above? 3(6xz-xy+3) 3(6z-y+3) 3x(6-y+3) 3x(6z-3y+x) 3x(6z-y+3)

E Correct. Don't worry about the expression containing three variables x, y and z - the extraction procedure is identical to earlier examples. Make sure the common factor you choose is indeed a factor of all the terms in the expression: the common factor needs to be the greatest number/variable combination that all of the terms are divisible by. Remember, you can check your work by expanding the brackets. First, find the greatest common factor of 18xz, 3xy and 9x. All terms are divisible by 3 and by x, so the common factor is 3x; Write it down outside a pair of brackets: 3x∙( ) Populate the brackets with the remaining factors. 3x must be multiplied by 6z to result in 18xz, by y to result in 3y, and by 3 to result in 9x. Remember to keep the original "+" and "-" signs: ---> 3x(6z-y+3)

Q4...If in the figure above, l1 and l2 are parallel, what is x? 105° 110° 120° 140° 150°

E Correct. Lines given in a parallel lines diagram can be extended to make it easier to solve the question. In this case, extend l2 backwards and then extend the top diagonal line to intersect both parallel lines. Extend l2 backwards and extend the top diagonal line so that it intersects both parallel lines. The resulting triangle is shown in the figure. The supplementary angle of 80 is 100 (180 - 80 = 100). Given that l1 and l2 are parallel lines, the alternate interior angle on l2 is also 50°. Since the sum of all angles in a triangle is 180, the supplementary angle of x is 30 (180 - 100 - 50 = 30). Hence, x = 180 - 30 = 150°.

The product of Ron's age, in years, and half of George's age, in years, is 42. If Ron is 5 years older than George is, then how old is Ron ? 7 8 9 10 12

E Correct. Numbers in the answer choices and a specific question ("How many years old?") call for Plugging In The Answers. You may feel like writing down one equation or more. This is just your algebraic urge, which is another stop sign for Reverse PI problems. Assume the amount in the answer choice is the age of Ron and then follow the story in the problem. If everything fits - stop. Pick it. Otherwise - POE and move on, until you find an answer that works. Start with answer choice C: Assume Ron is 9 years old. Then, George must be 4 years old. The product of Ron's age and half of Georges age, 9x2, is too small, so you can POE C. In which direction should you go? You need a bigger product, so their ages must be bigger. Therefore POE B and A. Now, plug in D or E to check which is correct. Plugging in E works: If Ron is 12, George is 7 and the product of Ron's age and half of George's age is 12x7/2=42.

Q8...AG, BF and CE are parallel. If the height of B from AG is 12, and the height of C from BF is 8, then what is the height of C from AG? 9 12 15 16 20

E Correct. Remember that height of one point from another is the perpendicular distance from the first to the second point. The height of point C from AG is the perpendicular distance of C from AG. Since AG and BF are parallel, every point on line BF is the same distance from line AG, and a height from any point on line BF to line AG will also equal 12. Thus, it is possible to draw a height from point C directly to line AG, and the resulting height will simply equal the sum of the heights from C to BF and from BF to AG, or 12 + 8 = 20. Hence, the height of C from line AG = 20.

If x=4 is a root of the equation x2=m−2x, what is the value of m? −24 −12 −4 12 24

E Correct. Remember the meaning of factoring solutions: The solution x1 is the value that satisfies the equation ax2+bx+c=0. Plugging it into the equation will give a result of zero. Change the quadratic equation into the form of ax2+bx+c=0: x2=m−2x --> x2+2·x-m=0 Plug x=4 into the quadratic equation x2+2·x-m=0 to form an easy equation with m: 42+2·4-m=0 Solve for m: --> 16+8=m --> m=24

If the supplement of a certain angle is six times the measure of the complement of that certain angle, then what is the measure of that certain angle? 36° 50° 56° 60° 72°

E Correct. Remember the supplement of an angle and the angle add up to 180°. Likewise, the complement of an angle and the angle add up to 90°. Translate English to Math. If the angle of the polygon is x, the supplement of x is (180 - x) and the complement of x is (90 - x). According to the question (180 - x) = 6 (90 - x) --> (180 - x) = 540 - 6x --> (6x - x) = 540 - 180 --> 5x = 360 --> x = 360/5 = 72. Hence, the angle is 72°. Alternatively, a specific question and numbers in the answer choices warrant Reverse PI. Start with the middle answer 56. Its complement is 34. If the supplement of the angle is 6 times as much it turns out that 34·6 is greater than 180. POE C, and also A and B, since the supplement-complement need be smaller. Now Reverse Plug in D or E to see which one is correct. The answer is E. The complement of 72 is 18, and 18·6=108, which is also the supplement.

Q1... Angles AMB and CMD are right angles, ∠AMD is acute, and ∠BMC=y° is acute. What is ∠AMD ? 90°−y 180°−y 90°+y 2y y

E Correct. Since there's no drawing, draw one yourself. Here are the two option for juxtaposing the two right angle ∠AMB and ∠CMD keeping both ∠AMD and ∠BMC acute. D is either to the left of M or to the right of M: In both cases the acute angle offsets right angle CMD by the equivalent of "y" degrees from right angle AMB. Therefore, ∠AMD=∠BMC=y°.

10512−10492=? 4 16 1050 2100 4200

E Correct. The difference between two squares is a dead giveaway for recycled quad III. --> 10512−10492= --> (1051-1049)×(1051+1049) = --> 2×2100 = 4200

Adam deposited a portion of his salary in a savings account in January 2005. Adam earns a 10% interest compounded annually. If Adam plans to make a withdrawal of all the money in the account in January 2010, then his withdrawal is approximately what percent of his initial deposit? 50% 61% 100% 150% 161%

E Correct. Whenever there's an invisible variable in the problem plug in a good number. If the problem asks about percents use 100 or multiples of it. In this case, plug in a good number for Adam's initial deposit. Remember that percent problems are a great opportunity to ballpark. Plug in 100 for Adam's initial deposit (the invisible variable) in 2005. He made 10% on the first year, 10% on the second year, and so forth. What keeps changing is the total worth of his deposit, which keeps on getting bigger every year. And so, 2005-2006: deposits 100, ends up with 100+10=110, 2006-2007: deposits 110, ends up with 110+11=121, 2007-2008: deposits 121, ends up with 121+12=133, 2008-2009: deposits 133, ends up with 133+13=146, 2009-2010: deposits 146, ends up with 146+15= 161. Assume that the interest is simple, i.e. 10% annually taken only from the original amount, instead of the successively growing balance. Thus, in 5 years, Adam will have a straightforward 150% of the original amount = 5*10%. Since the question actually uses "compound interest" the actual result must be higher than 150% because of the compounded interest. Only E fits that description, so it must be the right answer.

Q12....If pentagon ABCDE in the figure above is regular, what is x? 45° 36° 72° 90° 108°

E Good work! X and A are opposite angles and therefore equal to each other. To find the measure of X, find the measure of A. The pentagon ABCDE is regular, and therefore all of its sides and angles are equal. Since the sum of the angles in an n sided polygon is equal to 180°·(n-2), the sum of the angles in a pentagon is 540 (180·3). Since all the angles are equal, divide by 5 (the number of angles in the polygon) to find the measure of a single angle.

If x<y, and y(y−2x)=2−x2, what is the value of x−y? −2 −√2 −1 1 √2

E Incorrect. Expand the expression on the left hand side and look for familiar patterns. Note: x<y. B Correct. --> y(y−2x)=2−x2 --> y2-2xy=2-x2 This is starting to look familiar. Did anyone say Recycled Quad II? --> x2-2xy+y2=2 --> (x-y)2=2 Now this quadratic actually has two solutions: x-y=√2 or x-y=-√2. However, since x<y, extracting a square root will leave you with the negative root of 2: --> x-y=-√2

In the equation x2+14x+k=0, x is a variable, and k is a constant. If (x−1) is a factor of the equation x2+14x+k=0, then which of the following is a solution of x2+14x+k=0? −15 −14 −13 13 15 Alternative method: Remember: the solution for a quadratic equation of the form ax2+bx+c=0 is the value of x for which the quadratic equals zero. If (x-1) is a factor of the quadratic, then setting the factor to zero will yield x=1 as a solution of the quadratic. Plug in x=1 into the equation to find k, then factor the equation to find the other root.

E Incorrect. Recall the method for factoring a quadratic of the form ax2+bx+c=0: 1) Write in factorized form (x+e)(x+f)=0. Each of the pairs of parentheses is called a factor of the equation. 2) Find e and f: a pair of integers whose product is c and whose sum is b. 3) Write e and f in the appropriate places in the factors. e.g. (x+5)(x+3)=0 4) Set each factor to zero and solve. Since (x-1) is a factor of the quadratic, then e=-1. Find the corresponding f which fits the terms set in step 2 and proceed accordingly. Answer choice E is the value of f, the other factor of the quadratic. However, the question asked for the solution of the equation. A Correct. For the quadratic x2+14x+k=0, a=1; b=14; c=k; The sum of e and f must be equal to b=14. If e=-1 is one of the values of the factors, then -1+f=14, or f=15. Plug in e and f into the expanded form of the quadratic: (x-1)(x+15)=0. Set the factors to zero to find the solutions: x+15=0 --> x=-15

Which of the following describes all the values of x for which (x−1)2 > 1? −1 < x < 1 0 < x < 2 x < −2, or x > 0 x > 2, or x < 0 x > 2, or x < −2

E Incorrect. When x2 ≤ a, then -√a ≤ x ≤ √a. When x2 ≥ a, then x ≥ √a or x ≤ -√a. In this question you are dealing with the second scenario: If (x−1)2 > 1, then x-1 < -1 OR x-1 > 1. D Correct. x-1 < -1 --> x < 0 OR x-1 > 1 --> x > 2

The sum of the angles in an n sided polygon is given by the formula 180°·(n-2). Some examples: Since in a regular polygon of n sides , the sum of the angles is 180°·(n-2), each equal angle in a regular polygon is of measure 180°·(n-2)/n. For example, the sum of the angles in the regular pentagon is 180(5-2) = 180(3) = 540. Divide this by 5 angles to get 540/5 = 108° for each angle.

Name.....No of sides...Sum of ∠'s Triangle..............3...............180° Quadrilateral......4...............360° Pentagon............5...............540° Hexagon............6...............720° Octagon.............8.............1080° For n sided polygon: No of sides: n Sum of Angles: 180°×(n-2)

The height or altitude, is the vertical distance between two points. It is always perpendicular to a side or surface at one end, at least.

The height may be used in a polygon (a two dimensional figure with multiple sides), or in three-dimensional objects.

The distance from length to length is termed width. It is always at right angles to the length.

The width is measured in feet, inches, centimetres etc.

Each of the three angles in a triangle determines the distance between the rays that form the angle. In other words, the angle determines the size of the side opposite to it.

Therefore, the largest angle is opposite the largest side, the smallest angle is opposite the smallest side etc. In the same manner, the sides opposite equal angles are also equal, and vice versa.

A vertex (plural: vertices) is the intersection point of at least two straight lines, or rays (for angles). The term vertex often refers to a figure (i.e., triangle, square etc.), or an angle.

To bisect means to cut something exactly in half. A bisector is a line that cuts the angle in half. Note that when the GMAT uses 'bisector' to describe a line, it means just 'bisector of an angle'. Two lines (such as the diagonals of a rectangle) may bisect each other, but they are not considered 'bisectors'.

Triangle means three-angles. Every triangle is composed of three sides that form three angles. Sometimes one of the sides is referred to as the "base".

Triangles are perhaps the most common figure in GMAT Geometry problems. The GMAT tests your knowledge of triangles in terms of angles, lines, areas, and special triangles.

A straight angle measures exactly 180°. It is termed that way because of the straight line that determines it. A straight angle changes the direction to point the opposite way. Sometimes people would say: "you did a complete 180 on that matter...", meaning you turned your mind or direction the opposite way.

When two straight lines intersect, the angles opposite each other are equal. They are termed opposite angles or vertical angles (because they share the same vertex).

Parallel lines are straight lines that never meet. They are also termed equidistant, because their distance is always the same. Parallelism is marked by "||" (e.g., a||b)

When a third line intersects parallel lines, angles are formed. There are only two kinds of angles: big and small. Remember the rules regarding these angles; All the small angles equal each other. All the big angles equal each other. Any combination of Big angle+Small angle is always 180°.