GRE Geometry - Everything you need to know Geometry... · GRE Geometry - Everything you need to...

GRE Geometry - Everything you need to know

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This slideshow features screenshots from Greenlight Test Prep’s entire Geometry module (consisting of 42 videos). It covers every key concept you need to know about GRE Geometry. It also includes 32 practice questions.

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Lines and Angles

(watch the entire video here)

Lines and Angles

line: a straight path that extends without end in both directions

Lines and Angles

AB: line segment

AB: length of line segment AB (e.g., DE=7)

line: a straight path that extends without end in both directions

Lines and Angles

angle: intersection of 2 lines

: measured in degrees or radians

Lines and Angles

Angles on a line add to 180°

180a b c

70 180

Lines and Angles

right angle: angle of 90 degrees

PQ is perpendicular to AB

Lines and Angles

bisect: cut or divide into 2 equal pieces

JK bisects AB

bisect s ABC

bisector is the of ABC

line l is the perpendicular bisector of AB

Lines and Angles

- a and c are vertical angles

- a and c are opposite angles

- a and c are vertically opposite angles

- b and d are opposite angles

Opposite angles are equal

Aside: 180x y

Lines and Angles

50x 50 180

Lines and Angles

If two lines do not intersect, they are parallel

Lines and Angles

If two lines do not intersect, they are parallel

Note: 180x y

Lines and Angles

Practice Question

B) 17.5

E) 42.5

If l1 and l2 are parallel, then x =

Practice Question

B) 17.5

E) 42.5

If l1 and l2 are parallel, then x =

15x 3 5x

15 3 5 180

4 10 180

Triangles – Part I

B Cw x

y180w x y

Angles in a triangle add to 180°

44180w x y

B Cw x

The longest side is opposite the largest angle

The shortest side is opposite the smallest angle

If then a b c A B C

The sum of the lengths of any two sides of a triangle must be greater than the third side.

If a triangle has sides with lengths 3 and 7, what lengths are possible for the third side?

third side 73 37

rd difference between other 2 sides 3 side sum of other 2 sides

Given lengths of sides A and B

rd 3 sideA B A B

Given lengths of sides A and B

rd 3 sideA B A B

If then a b c A B C

The sum of the lengths of any two sides of a triangle must be greater than the third side.

Practice Question

Quantity A Quantity B

w, x and y are the lengths of the 3 sides of a certain triangle

2w x y

A: Quantity A is greater

B: Quantity B is greater

C: The two quantities are equal

D: The relationship cannot be determined

Practice Question

w, x and y are the lengths of the 3 sides of a certain triangle

2w x y

Given the lengths of sides A and B in a triangle:

rd 3 sideA B A B

ww x xy 2w wx x

Practice Question

What is the value of x in terms of y ?

Practice Question

43 180ya

43 22 180xa

43 22 43

Solution #1

Practice Question

158 18

22 180

158 180

Solution #2

Assumptions in Geometry

• Lines that appear straight can be assumed to be straight

• Do not make assumptions about angle measurements

• y +x =180

• Both angles are greater than zero degrees

• y is greater than x

• Do not make assumptions about parallelism

• Unless otherwise indicated, ABC refers to the smaller angle.

• Not necessarily drawn to scale

• Avoid visual estimation

Geometric Figures

• Angles are greater than 0 degrees

• Do not make assumptions about angle measurements

• Do not make assumptions about parallelism

• Unless otherwise indicated, ABC refers to the smaller angle

Geometry Strategies

• Redraw figures

• Add all given information

• Add all information that can be deduced

• Add/extend lines

• Assign variables and use algebra

• Drawn to scale estimate to confirm calculations and guide guesses

• Drawn to scale estimate measurements to confirm or guess

Triangles – Part II

Isosceles triangle

• 2 equal sides, 2 equal angles

If then a b c A B C

40 180

2 40 180

Equilateral triangle

• 3 equal sides, 3 equal angles

base heightArea

1Area base height

base heightArea

10Area

altitude height

base heightArea

C60 60

3 sideArea

The altitudes of isosceles triangles and equilateral triangles bisect the base.

• An isosceles triangle has 2 equal sides and 2 equal angles

• An equilateral triangle has 3 equal sides and 3 equal angles (60° each)

base heightArea

3 sideArea

• The altitudes of isosceles triangles and equilateral triangles bisect the base

Practice Question

A) 27.5

C) 62.5

E) 125

If AB and CD are parallel, and AB = BC, then x =

Practice Question

A) 27.5

C) 62.5

E) 125

If AB and CD are parallel, and AB = BC, then x =

110 18

Right Triangles

• Right triangle: triangle with right (90°) angle

• The hypotenuse is the longest side

1 2leg leg hypotenuse

2 2 2a b c

bc 2 2 2a b c

Right Triangles

2 2 2a b c

Right Triangles

2 2 2a b c

Right Triangles

2 2 2a b c

Right Triangles

• 3-4-5

• 5-12-13

• 8-15-17

2 2 23 4 5

2 2 25 12 13

• 7-24-25

Pythagorean triples: A set of 3 integers that can be the sides of a right triangle

Right Triangles

• 8-15-17

2 2 215 17x

2 2 2a b c

Right Triangles

• 3-4-5

• 5-12-13

• 8-15-17

• 7-24-25

6-8-10 9-12-15 12-16-20

10-24-26

4 7 28

5 7 35 213 7x

2 corresponding sides required to use Pythagorean triples

Right Triangles

• 3-4-5

• 5-12-13

• 8-15-17

• 7-24-25

6-8-10 9-12-15 12-16-20

10-24-26

Enlarged by factor

725 Enlarged

by factor of 2

2 corresponding sides required to use Pythagorean triples

Right Triangles

• 3-4-5

• 5-12-13

• 8-15-17

• 7-24-25

6-8-10 9-12-15 12-16-20

10-24-26

2 2 2a b c

Right Triangles

2 2 2a b c

• Watch out for Pythagorean triples (and their multiples)

5-12-13

8-15-17

7-24-25

Practice Question

AA) 2 3

B) 2 5

D) 4 3

E) 4 5

The height of this rectangle is twice its width. If the distance

between points A and B is , what is the rectangle’s height? 60

Practice Question

AA) 2 3

B) 2 5

D) 4 3

E) 4 5

2 2 2a b c

2 2 3x

The height of this rectangle is twice its width. If the distance

between points A and B is , what is the rectangle’s height? 60

Practice Question

C) 2 21

D) 149

E) 3 21

If the rectangular box shown here is 6 inches wide, 8 inches long and 7

inches high, what is the distance, in inches, between points A and B ?

Practice Question

C) 2 21

D) 149

E) 3 21

2 2 2a b c 2 2 2

100 49

Solution #1

Practice Question

C) 2 21

D) 149

E) 3 21

2 2 2AB w x y

2 2 28 6 7

64 36 49

Solution #2

Special Right Triangles

45-45-90 triangle

2 1.445

leg : leg : hypotenuse

30-60-90 triangle

leg : leg : hypotenuse

1 : : 2

3xx : : 2x

enlargement factor: 6

enlargement factor:

Square Equilateral Triangle

Watch out for special right triangles “hiding” in squares and equilateral triangles

Practice Question

A) 3 2

B) 2 6

C) 4 3

D) 6 2

E) 6 3 B

If , 6 and 105 , then AD BD AB ABC x

Note: Figure not drawn to scale

Practice Question

A) 3 2

B) 2 6

C) 4 3

D) 6 2

E) 6 3 B

If , 6 and 105 , then AD BD AB ABC x

enlargement factor: ? 6

Similar Triangles

Similar triangles have the same 3 angles in common

40 20120

1203 30 1 55.6 60x

Similar Triangles

40 20120

With similar triangles, the ratio of any pair of corresponding sides is the same

b c xy

Similar Triangles

40 20120

b c xy

Practice Question

If , then ABC BCD x

Practice Question

If , then ABC BCD x

❤ With similar triangles, the ratio of any pair

of corresponding sides is the same

Quadrilaterals

Angles in a quadrilateral add to 360°

y360w x y z

Quadrilaterals

square

rectangle

trapezoid

parallelogram

rhombus

Quadrilaterals

parallelogram

opposite sides parallel

rectangle

all angles are 90

rhombus

all sides are equal

square

Quadrilaterals

trapezoid

2 sides parallel

Quadrilaterals

Rhombus (and square)

• diagonals are perpendicular bisectors

Rectangle (and square)

• diagonals are equal length

Quadrilaterals

square rectangle

trapezoid

area base height

base base

height height

height

1 2base basearea height

average of bases height

parallelogram rhombus

height

Quadrilaterals

rhombus

1 2diagonal diagonalarea

Quadrilaterals

Angles in a quadrilateral add to 360°

parallelogram

rectangle

all angles are 90

rhombus

all sides are equal

square

trapezoid

2 sides parallel

area base height

Polygons

Polygon: Closed figure formed by 3 or more line segments

Polygons

“polygon” “convex polygon” (all interior angles less than 180°)

Polygons

180a b c

Triangle

Quadrilateral

Pentagon

d360a b c d

cd 540a b c d e

Hexagon

cd 720a b c d e f

Polygons

The sum of the interior

angles in an N-sided polygon

is equal to 180 2N

Octagon

sum of angles 180 2

8180 2

Polygons

Regular polygon: equal sides and equal angles

regular pentagon

Polygons

• Polygon: Closed figure formed by 3 or more line segments

• “polygon” “convex polygon” (all interior angles less than 180°)

Triangle Quadrilateral

Pentagon Hexagon

• Regular polygon: equal sides and equal angles

is equal to 180 2N

Circles

Circle: set of points that are equidistant from a given point

center

diameter u2 radi s

- “arc CDE ”

- “minor arc CE ”

Circles

circumference 2 radius

Circumference 3.14

circumference diameter

Circles

circumference 2 r

Circumference

circumference 2

16 feet

Circles

2area r

Circles

center

circumference diameter

circumference 2 radius

2area r

Practice Question

The area of circle A is twice the area of circle B.

The radius of circle A The diameter of circle B

Practice Question

Quantity A Quantity B B

The area of circle A is twice the area of circle B.

circle A's radiusa

The radius of circle A The diameter of circle B

circle B's radiusb

circle A area 2 circle B area

2area r

divide by b

Practice Question

If is the center, 45 , and 6,then the area of the circle isO OBC BC

Practice Question

If is the center, 45 , and 6,then the area of the circle isO OBC BC

With similar triangles, the ratio of any pair of corresponding

sides is the same

2area r

Pieces of Pi

1of circumference

90of circumference

Pieces of Pi

119of circumference

Pieces of Pi

of circumference360

arc length 2360

Pieces of Pi

of area circ of sect le's area3

2sector area360

Pieces of Pi

2area360

26area360

Pieces of Pi

2area360

Practice Question

O is the center of the circle with radius 30. If x – w=20, what is the length of arc CDE ?

Practice Question

O is the center of the circle with radius 30. If x – w=20, what is the length of arc CDE ?

arc length 2360

180x w

arc length 2360

Circle Properties

“x is an inscribed angle holding/containing chord AB ”

“x is an inscribed angle holding/containing arc AB ”

Circle Properties

Inscribed angles holding the same chord/arc are equal

Circle Properties

Inscribed angles holding chords/arcs of equal length are equal

Circle Properties

An inscribed angle holding the diameter is a right angle

Circle Properties

“Angle AOB is a central angle holding chord AB”

A central angle is twice as large as an inscribed angle holding the same chord/arc

Circle Properties

The line from the center to the point of tangency is

perpendicular to the tangent line

“line l is tangent to the circle”

Circle Properties

Practice Question

If is the center and , then O AB CD x

Practice Question

If is the center and , then O AB CD x

Volume & Surface Area

1 ft1 ft

1 ft31 ft

2 ft3 ft

Volume length width height

Volume 2 3 5

Volume

height h

2Volume r h

3Vo 1lume

Volume

Surface Area

• 6 faces

• 12 edges

• 8 vertices

Surface Area

• 6 faces

• 12 edges

• 8 vertices

Surface Area

• 6 faces

• 12 edges

• 8 vertices

vertex

Surface Area

surface area 40 40 32 32 20 20

184 cm

Surface Area

surface area 40 40 32 32 20 20

184 cm

Surface Area

2area r 2area r

area 2

total area 2

r r rh

length

volume length width height

height

2volume r h

surface area 2

r r rh

surface area sum of areas of all 6 sides

Units of Measurement

• If conversion is required, relationship will be given

- e.g., (1 kilometer = 1000 meters)

- e.g., (1 mile = 5280 feet)

• Note: Relationships not given for units of time

- e.g., (1 hour = 60 minutes)

Conversions

- e.g., (1 day = 24 hours)

Geometry Strategies – Part II

• Redraw figures

• Add all given information

• Add any information that can be deduced

• Add/extend lines

• Assign variables and use algebra

• Problem solving questions drawn to scale:

• Circle:

• Break areas/volumes into manageable pieces

• Two or more triangles and length required

• Right triangle:

- use Pythagorean Theorem to relate sides

- watch for Pythagorean Triples and special triangles

- beware of circle properties (inscribed/central angles, tangent lines)

- look for isosceles triangles

- estimate to confirm calculations and guide guesses

- look for similar triangles

(watch the entire video here) Geometry Strategies – Part II

Practice Question

Area of an equilateral triangle with each side

having length 4k

Area of a circle with diameter 2k

Practice Question

3 sidearea

Area of an equilateral triangle with each side

having length 4k

Area of a circle with diameter 2k

2area r

(radius = k)

4 32divide by k

Practice Question

1x 4 3x

What is the value of x ?

Practice Question

1x 4 3x

What is the value of x ?

With similar triangles, the ratio of any pair of corresponding

sides is the same

2 1 1 4 3

2 2 4 3

Practice Question

Perimeter of ABC 10

ABC is an isosceles triangle

Practice Question

Quantity A Quantity B D

Perimeter of ABC 10

ABC is an isosceles triangle

3BC 10 10=

4BC 11 10

Practice Question

If is tangent to the circle with center , then AC O DBC

A) 50°

B) 55°

C) 60°

D) 65°

E) 70°

Practice Question

If is tangent to the circle with center , then AC O DBC

A) 50°

B) 55°

C) 60°

D) 65°

E) 70°

Practice Question

5x2x y

Practice Question

5 2 90 180

7 0 18

5x2x y

Practice Question

What is the area of triangle ?ABC

B) 15 3

5 119C)

D) 32.5

Practice Question

What is the area of triangle ?ABC

B) 15 3

5 119C)

D) 32.5

3 6 36h

base heightarea

5 6 3area

Which of the following could equal the length of side ?KM

Indicate all possible values.

(watch the entire video here) Practice Question

Which of the following could equal the length of side ?KM

Indicate all possible values.

Given lengths of sides A and B in a triangle:

rd 3 sideA B A B

4 2 4 2

(watch the entire video here) Practice Question

Practice Question

Volume of spherewith radius 2

Practice Question

Volume of spherewith radius 2

sphere volume

sphere

cube volume

4 volume

sphere volu

If is a parallelogram, then what is its perimeter?ABCD

3 3x y

4 2 2y x

6x y 2 6 13x y

Practice Question

If is a parallelogram, then what is its perimeter?ABCD

perimeter 3 3 2 6 13 4 2 2

4 4 18

x y x y y x x y

6 2 6 13

x y x y

6x y 2 6 13x y 3 3 4 2 2

x y y x

4 2 2y x

3 3x y

Practice Question

In the regular octagon shown here, what is the value of x?

x degrees

Practice Question

is equal to 180 2N

In the regular octagon shown here, what is the value of x?

x 135 degrees

sum of all angl8 8

es 180 2

180 6eac

6h angle 1

Practice Question

The diagonal of square has length 3.23ABCD

Area of square ABCD 3.23

Practice Question

The diagonal of square has length 3.23ABCD

Area of square ABCD 3.23

Quantity A Quantity B B

3.23 2 2 2a b c

2 3.23

Practice Question

What is the value of ?x

155 3x

Practice Question

What is the value of ?x

155 3x

180 4 70x

180 155 3x

180 4 70 180 155 3 6 30 180

110 4 25 3 6 30 180

105 5 180

Practice Question

The above circle is tangent to the x- and y-axes. It has radius 4, and its center is at O.

's distance from the originO

Practice Question

The above circle is tangent to the x- and y-axes. It has radius 4, and its center is at O.

's distance from the originO

Practice Question

K is the surface area of cylinder A. If the radius of cylinder B is twice the radius of cylinder A, and the height of cylinder B is twice that of cylinder A, what is the surface area of cylinder B?

Practice Question

K is the surface area of cylinder A. If the radius of cylinder B is twice the radius of cylinder A, and the height of cylinder B is twice that of cylinder A, what is the surface area of cylinder B?

2surface area 2 2r rh 1

2surface area 2 2r rh

2 14 48 0y y

Area of .ABC 12

Practice Question

2 14 48 0y y

Area of .ABC 12

2 14 48 0

6 , 80 68

8 3ar 1ea

12= 12

Practice Question (watch the entire video here)

Practice Question

If bisects , and bisects , then BD CBE DE BEF w

Practice Question

If bisects , and bisects , then BD CBE DE BEF w

yy 180 2y

180 2x

50 180 2 180 2 180

410 2 2 180

230 2 2

Practice Question

If is a rectangle, then what is the length of ?ABCD EC

A) 7.8

C) 8.4

E) 9.6

Practice Question

If is a rectangle, then what is the length of ?ABCD EC

A) 7.8

C) 8.4

E) 9.6

121216

12 16area

Practice Question

Area of .ABC 6

In the rectangular coordinate plane, ABC has vertices at (0,-2), (0,1) and (4,b), where b<0.

Practice Question

Area of .ABC 6

In the rectangular coordinate plane, ABC has vertices at (0,-2), (0,1) and (4,b), where b<0.

base heightArea

Practice Question

If the both circles have radius 6, and O and P are their centers, what is the area of the shaded region?

A) 24 18 3

B) 24 12 3

D) 36 24 3

E) 18 12 3

Practice Question

If the both circles have radius 6, and O and P are their centers, what is the area of the shaded region?

A) 24 18 3

B) 24 12 3

D) 36 24 3

E) 18 12 3

2sector area360

b d b d

a b d e b c d f

a b c d e f

3 sidearea

b d a b c d e f

For additional practice questions, see the bottom of our Geometry module

If you enjoyed this unique learning format, let us know, and we’ll add similar

resources to our SlideShare page

GRE Geometry - Everything you need to know Geometry... · GRE Geometry - Everything you need to...

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