Euclidean Geometry Cheat Sheet

Struggling to remember the exact wording for geometry proofs? This tool is designed to help HKDSE Mathematics students instantly find and review geometry reasons covered in the syllabus.

Lines

Triangles

Quadrilaterals

Circles

Let O be the centre of all the circles below.

Tangents

Euclidean Geometry Cheat Sheet

Struggling to remember the exact wording for geometry proofs? This tool is designed to help HKDSE Mathematics students instantly find and review geometry reasons covered in the syllabus.

Lines

adj. $∠$ s on st. line

$adj. $\angle$s on st. line$

If AOB is a straight line,

Then

x + y = 18 0^{\circ}

adj. $∠$ s supp.

$adj. $\angle$s supp.$

x + y = 18 0^{\circ}

Then

AOB is a straight line.

$∠$ s at a pt.

$$\angle$s at a pt.$

w + x + y + z = 36 0^{\circ}

vert. opp. $∠$ s

$vert. opp. $\angle$s$

If lines AB and CD meet at O,

Then

x = y

corr. $∠$ s, AB $∥$ CD

$corr. $\angle$s, AB$\parallel$CD$

A B ∥ C D

Then

x = y

alt. $∠$ s, AB $∥$ CD

$alt. $\angle$s, AB$\parallel$CD$

A B ∥ C D

Then

x = y

int. $∠$ s, AB $∥$ CD

$int. $\angle$s, AB$\parallel$CD$

A B ∥ C D

Then

x + y = 18 0^{\circ}

corr. $∠$ s equal

$corr. $\angle$s equal$

x = y

Then

A B ∥ C D

alt. $∠$ s equal

$alt. $\angle$s equal$

x = y

Then

A B ∥ C D

int. $∠$ s supp.

$int. $\angle$s supp.$

x + y = 18 0^{\circ}

Then

A B ∥ C D

mid-pt. theorem

A D = D B

and

A E = E C

(D,E midpoints of AB, AC),

Then

D E ∥ B C

B C = 2 D E

intercept theorem

A D ∥ B E ∥ C F

and

A B = B C

Then

D E = E F

Triangles

$∠$ sum of $△$

$\angle$ sum of $\triangle$

x + y + z = 18 0^{\circ}

ext. $∠$ of $△$

$ext. $\angle$ of $\triangle$$

x + y = z

SAS

A B = P Q

∠ A B C = ∠ P QR

B C = QR

Then

△ A B C ≅ △ P QR

SSS

A B = P Q

B C = QR

A C = P R

Then

△ A B C ≅ △ P QR

ASA

∠ A B C = ∠ P QR

B C = QR

∠ A C B = ∠ P R Q

Then

△ A B C ≅ △ P QR

AAS

∠ A B C = ∠ P QR

∠ A C B = ∠ P R Q

A B = P Q

Then

△ A B C ≅ △ P QR

RHS

∠ A B C = ∠ P QR = 9 0^{\circ}

A C = P R

(hypotenuse),

B C = QR

(side),

Then

△ A B C ≅ △ P QR

AAA

a = p

b = q

c = r

Then

△ A B C ≅ △ P QR

AA

b = q

c = r

Then

△ A B C \sim △ P QR

3 sides proportional

\frac{A B}{P Q} = \frac{B C}{QR} = \frac{A C}{P R}

Then

△ A B C \sim △ P QR

ratio of 2 sides, inc. $∠$

$ratio of 2 sides, inc. $\angle$$

b = q

and

\frac{A B}{P Q} = \frac{B C}{QR}

Then

△ A B C \sim △ P QR

corr. $∠$ s, $≅$ $△$ s

$corr. $\angle$s, $\cong$ $\triangle$s$

△ A B C ≅ △ P QR

Then

∠ A B C = ∠ P QR

∠ B C A = ∠ QR P

∠ A C B = ∠ P R Q

corr. sides, $≅$ $△$ s

$corr. sides, $\cong$ $\triangle$s$

△ A B C ≅ △ P QR

Then

A B = P Q

B C = QR

A C = P R

corr. $∠$ s, $\sim$ $△$ s

$corr. $\angle$s, $\sim$ $\triangle$s$

△ A B C \sim △ P QR

Then

∠ A B C = ∠ P QR

∠ B C A = ∠ QR P

∠ A C B = ∠ P R Q

corr. sides, $\sim$ $△$ s

$corr. sides, $\sim$ $\triangle$s$

△ A B C \sim △ P QR

Then

\frac{A B}{P Q} = \frac{B C}{QR} = \frac{A C}{P R}

base $∠$ s, isos. $△$

$base $\angle$s, isos. $\triangle$$

A B = B C

Then

∠ A B C = ∠ A C B

sides opp. equal $∠$ s

$sides opp. equal $\angle$s$

∠ A B C = ∠ A C B

Then

A B = A C

property of isos. $△$

$property of isos. $\triangle$$

△ A B C

is isosceles (

A B = A C

Then

∠ B A D = ∠ C A D

B D = C D

A D ⊥ B C

property of equil. $△$

$property of equil. $\triangle$$

△ A B C

is equilateral (

A B = B C = A C

Then

∠ A B C = ∠ A C B = ∠ B A C = 6 0^{\circ}

pyth. thm

△ A B C

is right-angled (at B),

Then

A B^{2} + B C^{2} = A C^{2}

converse of pyth. thm

A B^{2} + B C^{2} = A C^{2}

Then

△ A B C

is right-angled and

∠ A B C = 9 0^{\circ}

triangle inequality

A B + A C > B C

A B + B C > A C

A C + B C > A B

Quadrilaterals

$∠$ sum of polygon

$$\angle$ sum of polygon$

Sum of interior angles = (n - 2) \times 18 0^{\circ}

e.g. sum of interior angles of pentagon

(5 - 2) \times 18 0^{\circ} = 54 0^{\circ}

a + b + c + d + e = 54 0^{\circ}

sum of ext. $∠$ s of polygon

$sum of ext. $\angle$s of polygon$

Sum of exterior angles = 36 0^{\circ}

e.g.

a + b + c + d + e = 36 0^{\circ}

opp. sides of $∥$ gram

$opp. sides of $\parallel$gram$

A B C D

is a parallelogram,

Then

A B = D C

A D = B C

opp. $∠$ s of $∥$ gram

$opp. $\angle$s of $\parallel$gram$

A B C D

is a parallelogram,

Then

∠ D A B = ∠ B C D

∠ A D B = ∠ A B C

diags. of $∥$ gram

$diags. of $\parallel$gram$

A B C D

is a parallelogram,

Then

O D = O B

O A = O C

diagonals bisect area of $∥$ gram

$diagonals bisect area of $\parallel$gram$

A B C D

is a parallelogram,

Then

Area of △ A C D = Area of △ A C B

Area of △ A B D = Area of △ C B D

opp. sides equal

A B = D C

and

A D = B C

Then

A B C D

is a parallelogram

opp. $∠$ s equal

$opp. $\angle$s equal$

∠ A D C = ∠ A B C

and

∠ D A B = ∠ D C B

Then

A B C D

is a parallelogram

diags. bisect each other

O A = O C

and

O D = O B

Then

A B C D

is a parallelogram

opp. sides equal and $∥$

$opp. sides equal and $\parallel$$

A B = D C

and

D C ∥ A B

Then

A B C D

is a parallelogram

property of square

A B C D

is a square,

Then

A C = B D

A C ⊥ B D

∠ A D C = ∠ D C B = ∠ C B A = ∠ B A D = 9 0^{\circ}

A B = B C = C D = D A

D B

bisects

∠ A B C

and

∠ A D C

A C

bisects

∠ D A B

and

∠ D C B

it posses all properties of a parallelogram

property of rectangle

A B C D

is a rectangle,

Then

A C = B D

∠ A D C = ∠ D C B = ∠ C B A = ∠ B A D = 9 0^{\circ}

it posses all properties of a parallelogram

property of rhombus

A B C D

is a rhombus,

Then

A B = B C = C D = D A

A C ⊥ B D

A C

bisects

∠ B A D

and

∠ B C D

B D

bisects

∠ A B C

and

∠ A D C

it posses all properties of a parallelogram

property of kite

A B C D

is a kite,

Then

A C ⊥ B D

A D = D B

D C = B C

O D = O B

∠ D A O = ∠ B A O

∠ D C O = ∠ B C O

property of trapezium

A B C D

is a trapezium,

Then

D C ∥ A B

∠ C D A + ∠ D A B = 18 0^{\circ}

∠ D C B + ∠ C B A = 18 0^{\circ}

Circles

Let O be the centre of all the circles below.

line from centre $⊥$ chord bisects chord

$line from centre $\perp$ chord bisects chord$

O N ⊥ A B

Then

A N = N B

line joining centre to mid-pt. of chord $⊥$ chord

$line joining centre to mid-pt. of chord $\perp$ chord$

A N = N B

Then

O N ⊥ A B

(

∠ A N O

and

∠ O N B = 9 0^{\circ}

)

$⊥$ bisector of chord passes through center

$$\perp$ bisector of chord passes through center$

A N = N B

and

C D ⊥ A B

Then

C D

passes through the center O

equal chords, equidistant from centre

A B = C D

and

O M ⊥ A B

and

O N ⊥ C D

Then

O M = O N

chords equidistant from centre are equal

O M = O N

and

O M ⊥ A B

and

O N ⊥ C D

Then

A B = C D

$∠$ at center twice $∠$ at ☉ $^{ce}$

$\angle$ at center twice $\angle$ at ☉$^{ce}$

∠ A O B = 2 ∠ A C B

$∠$ in semi-circle

$$\angle$ in semi-circle$

If AB is the diameter of the circle,

Then

∠ A C B = 9 0^{\circ}

$∠$ s in the same segment

$$\angle$s in the same segment$

∠ A P B = ∠ A QB

arcs proportional to $∠$ s at centre

$arcs proportional to $\angle$s at centre$

A B ⌢ : B C ⌢ = ∠ A O B : ∠ B O C

arcs proportional to $∠$ s at circumference

$arcs proportional to $\angle$s at circumference$

A B ⌢ : B C ⌢ = ∠ A P B : ∠ B P C

equal chords, equal arcs

A B = C D

Then

A B ⌢ = C D ⌢

equal arcs, equal chords

A B ⌢ = C D ⌢

Then

A B = C D

equal arcs, equal $∠$ s

$equal arcs, equal $\angle$s$

A B ⌢ = C D ⌢

Then

∠ A O B = ∠ C O D

equal $∠$ s, equal arcs

$equal $\angle$s, equal arcs$

∠ A O B = ∠ C O D

Then

A B ⌢ = C D ⌢

equal chords, equal $∠$ s

$equal chords, equal $\angle$s$

A B = C D

Then

∠ A O B = ∠ C O D

equal $∠$ s, equal chords

$equal $\angle$s, equal chords$

∠ A O B = ∠ C O D

Then

A B = C D

opp. $∠$ s, cyclic quad.

$opp. $\angle$s, cyclic quad.$

A B C D

is a cyclic quadrilateral,

Then

∠ A B C + ∠ A D C = 18 0^{\circ}

∠ B A D + ∠ B C D = 18 0^{\circ}

ext. $∠$ s, cyclic quad.

$ext. $\angle$s, cyclic quad.$

A B C D

is a cyclic quadrilateral,

Then

∠ A B C = ∠ A D E

converse of $∠$ s in the same segment

$converse of $\angle$s in the same segment$

∠ A P B = ∠ A QB

Then

A,B,Q,P are concyclic

opp. $∠$ s supp.

$opp. $\angle$s supp.$

∠ D A B + ∠ B C D = 18 0^{\circ}

Then

A B C D

are concyclic

ext. $∠$ = int. opp. $∠$

$ext. $\angle$ = int. opp. $\angle$$

∠ A D C = ∠ C B E

Then

A B C D

are concyclic

Tangents

tangent $⊥$ radius

$tangent $\perp$ radius$

If PQ is tangent at T,

Then

P Q ⊥ O T

converse of tangent $⊥$ radius

$converse of tangent $\perp$ radius$

O T ⊥ P Q

Then

P Q

is tangent at T

tangent properties

If TP and TQ are tangents at P and Q from T,

Then

∠ T O P = ∠ T O Q

∠ O T P = ∠ O T Q

T P = T Q

$∠$ s in alt. segment

$$\angle$s in alt. segment$

If PQ is tangent at A,

Then

∠ Q A C = ∠ A B C

converse of $∠$ s in alt. segment

$converse of $\angle$s in alt. segment$

∠ Q A C = ∠ A B C

Then

P Q

is tangent at A

Lines

Triangles

Quadrilaterals

Circles

Tangents

Lines

adj. ∠s on st. line

adj. ∠s supp.

∠s at a pt.

vert. opp. ∠s

corr. ∠s, AB∥CD

alt. ∠s, AB∥CD

int. ∠s, AB∥CD

corr. ∠s equal

alt. ∠s equal

int. ∠s supp.

mid-pt. theorem

intercept theorem

Triangles

∠ sum of △

ext. ∠ of △

SAS

SSS

ASA

AAS

RHS

AAA

AA

3 sides proportional

ratio of 2 sides, inc. ∠

corr. ∠s, ≅ △s

corr. sides, ≅ △s

corr. ∠s, ∼ △s

corr. sides, ∼ △s

base ∠s, isos. △

sides opp. equal ∠s

property of isos. △

property of equil. △

pyth. thm

converse of pyth. thm

triangle inequality

Quadrilaterals

∠ sum of polygon

sum of ext. ∠s of polygon

opp. sides of ∥gram

opp. ∠s of ∥gram

diags. of ∥gram

diagonals bisect area of ∥gram

opp. sides equal

opp. ∠s equal

diags. bisect each other

opp. sides equal and ∥

property of square

property of rectangle

property of rhombus

property of kite

property of trapezium

Circles

line from centre ⊥ chord bisects chord

line joining centre to mid-pt. of chord ⊥ chord

⊥ bisector of chord passes through center

equal chords, equidistant from centre

chords equidistant from centre are equal

∠ at center twice ∠ at ☉ce

∠ in semi-circle

∠s in the same segment

arcs proportional to ∠s at centre

arcs proportional to ∠s at circumference

equal chords, equal arcs

equal arcs, equal chords

equal arcs, equal ∠s

equal ∠s, equal arcs

equal chords, equal ∠s

equal ∠s, equal chords

opp. ∠s, cyclic quad.

ext. ∠s, cyclic quad.

converse of ∠s in the same segment

opp. ∠s supp.

ext. ∠ = int. opp. ∠

Tangents

adj. $∠$ s on st. line

adj. $∠$ s supp.

$∠$ s at a pt.

vert. opp. $∠$ s

corr. $∠$ s, AB $∥$ CD

alt. $∠$ s, AB $∥$ CD

int. $∠$ s, AB $∥$ CD

corr. $∠$ s equal

alt. $∠$ s equal

int. $∠$ s supp.

$∠$ sum of $△$

ext. $∠$ of $△$

ratio of 2 sides, inc. $∠$

corr. $∠$ s, $≅$ $△$ s

corr. sides, $≅$ $△$ s

corr. $∠$ s, $\sim$ $△$ s

corr. sides, $\sim$ $△$ s

base $∠$ s, isos. $△$

sides opp. equal $∠$ s

property of isos. $△$

property of equil. $△$

$∠$ sum of polygon

sum of ext. $∠$ s of polygon

opp. sides of $∥$ gram

opp. $∠$ s of $∥$ gram

diags. of $∥$ gram

diagonals bisect area of $∥$ gram

opp. $∠$ s equal

opp. sides equal and $∥$

line from centre $⊥$ chord bisects chord

line joining centre to mid-pt. of chord $⊥$ chord

$⊥$ bisector of chord passes through center

$∠$ at center twice $∠$ at ☉ $^{ce}$

$∠$ in semi-circle

$∠$ s in the same segment

arcs proportional to $∠$ s at centre

arcs proportional to $∠$ s at circumference

equal arcs, equal $∠$ s

equal $∠$ s, equal arcs

equal chords, equal $∠$ s

equal $∠$ s, equal chords

opp. $∠$ s, cyclic quad.

ext. $∠$ s, cyclic quad.

converse of $∠$ s in the same segment

opp. $∠$ s supp.

ext. $∠$ = int. opp. $∠$

tangent $⊥$ radius

converse of tangent $⊥$ radius

$∠$ s in alt. segment

converse of $∠$ s in alt. segment

adj. $∠$ s on st. line

adj. $∠$ s supp.

$∠$ s at a pt.

vert. opp. $∠$ s

corr. $∠$ s, AB $∥$ CD

alt. $∠$ s, AB $∥$ CD

int. $∠$ s, AB $∥$ CD

corr. $∠$ s equal

alt. $∠$ s equal

int. $∠$ s supp.

$∠$ sum of $△$

ext. $∠$ of $△$

ratio of 2 sides, inc. $∠$

corr. $∠$ s, $≅$ $△$ s

corr. sides, $≅$ $△$ s

corr. $∠$ s, $\sim$ $△$ s

corr. sides, $\sim$ $△$ s

base $∠$ s, isos. $△$

sides opp. equal $∠$ s

property of isos. $△$

property of equil. $△$

$∠$ sum of polygon

sum of ext. $∠$ s of polygon

opp. sides of $∥$ gram

opp. $∠$ s of $∥$ gram

diags. of $∥$ gram

diagonals bisect area of $∥$ gram

opp. $∠$ s equal

opp. sides equal and $∥$

line from centre $⊥$ chord bisects chord