Geometry theorems and proofs summary

GEOMETRY THEOREMS AND PROOFS

The policy of the HAHS Mathematics staff when teaching Geometry Proofs is to have students present a solution in which there is a full equation showing the geometric property that is being used and a

worded reason that again identifies the geometric property that is being used. EXAMPLE:

Find the value of x.

C B A x 42 73

EQUATION REASON COMMENT

65 180 115 180 73 42       x x x (Angle sum of  ABCequals180 )

Desired level of proof to be reproduced by students – full equation contains geometric property and reason contains geometric property

General Notes:

(1) the word “equals” may be replaced by the symbol “=” or words such as “is”

(2) abbreviations such as “coint”, “alt”, “vert opp”, etc are not to be used – words are to be written in full

(3) the angle symbol (), the triangle symbol (), the parallel symbol (||), the perpendicular symbol (), etc are not to be used as substitutes for words unless used with labels such as PQR,

ABC, AB||XY, PQST

(4) If the geometric shape is not labelled then the students may introduce their own labels or refer to the shape in general terms such as “angle sum of triangle = 180o_{” or “angle sum of straight}

Revolution, Straight Angles, Adjacent angles, Vertically opposite angles

The sum of angles about a point is 360o. (angles in a revolution)

Find the value of x.

P 165 60 x 2x D C B A 360 165 60

2xx   (angle sum at a point P

equals 360o) 360 225 3x  135 3x 45  x

A right angle equals 90o.

AB is perpendicular to BC. Find the value of x.

D C B A x 36 90 36 

x (angle sum of right angle ABC

equals 90o) 54



x

A straight angle equals 180o.

FMJ is a straight segment. Find the value of x.

J I H G F M 50 46 4x 2x 180 50 46 4

2x x   (angle sum of straight

angle FMJ equals 180o) 180 96 6x  84 6x 14  x

Three points are collinear if they form a straight angle

Given that AKB is a straight line.

Prove that the points P, K and Q are collinear.

Q P K B A 72 3x 2x 180 2

3x x (angle sum of straight angle AKB equals 180o) 180 5x 36  x









 P, K and Q are collinear (PKQ is a straight

angle) * * PKQ equals 180o

Vertically opposite angles are equal.

AC and DE are straight lines. Find the value of y.

y 29 D B E C A 67 67 29 

y (vertically opposite angles are equal) 38



Angles and Parallel Lines

Alternate angles on parallel lines are equal.

All lines are straight. Find the value of x.

>>

>>

x (alternate angles are equal, AB||CD)

Corresponding angles on parallel lines are equal.

All lines are straight. Find the value of x.

>>

>>

x (corresponding angles are equal, AB||CD)

Cointerior angles on parallel lines are supplementary.

All lines are straight. Find the value of x.

>>

>>

x (cointerior angles are

supplementary, AB||CD) 55



Two lines are parallel if a pair of alternate angles are equal Prove that AB // CD 73 73 H G C D A B E F GHD AGH   (both 73o) ** CD AB ||

 (alternate angles are equal)

** equality of the angles involved must be clearly indicated

Two lines are parallel if a pair of corresponding angles are equal

Prove that AB // CD 65 65 H G C D A B E F EGB = GHD (both 65o) ** CD AB ||

 (corresponding angles are equal)

** equality of the angles involved must be clearly indicated

Two lines are parallel if a pair of cointerior angles are supplementary

Prove that PR // KM 56 124 L Q K M P R X Y RQL + QLM = 124o + 56o ** = 180o KM PR ||

 (cointerior angles are

supplementary)

* RQL + QLR = 180o

** supplementary nature of the angles involved must be clearly indicated

Angles in Polygons

The angle sum of a triangle is 180o.

Find the value of x.

A B C x 34 67 o o o 180 34 67  

x (angle sum of ABC equals

180o) 180 101  x 79  x

The exterior angle of a triangle equals the sum of the opposite (or remote) interior angles.

Find the value of x.

A C D B x 47 68o o o 47 68 

x (exterior angle of ABC equals sum

of the two opposite interior angles) 115



x

* exterior angle of ABC equals sum of remote interior angles

The angle sum of the exterior angles of a triangle is 360o.

Find the value of x.

A _C B x 157 128o o o 360 128 157  

x (sum of exterior angles of

ABC  equals 360o) 360 285  x 75  x

The angles opposite equal sides of a triangle are equal. (converse is true)

Find the value of x.

||

=

x (equal angles are opposite equal sides in

ABC

 ) *

The sides opposite equal angles of a triangle are equal (converse is true).

Find the value of x.

12 15 x A B C 65o 65o 15 

x (equal sides are opposite equal angles in

ABC

 )

All angles at the vertices of an equilateral triangle are 60o.

ABC

 is equilateral. EC and DB are angle bisectors and meet at P. Find the size of CPB.

B C

A

P

D E

ACB = 60o (all angles of an equilateral triangle are 60o)

similarly ABC = 60o

ECB = 30o (EC bisects ACB)

similarly DBC = 30o

CPB + 60o = 180o (angle sum of PCB equals 180o)

CPB = 120o

The angle sum of a quadrilateral is 360o.

Find the value of x.

A B C D o o o o x 3x 130 70 360 200

4x  (angle sum of quadrilateral ABCD

equals 360o) 40 160 4   x x

The angle sum of a n-sided polygon is 180(n – 2)o or (2n – 4) right angles.

Find the value of x.

106 87 x 165 92 B C D E A

Angle sum of a pentagon = 3  180o = 540o

x + 450 = 540 (angle sum of pentagon equals

540o)

x = 90

The angle at each vertex of a regular n-sided polygon is





o 2 180      n n .

Find the size of each interior angle of a regular hexagon            120 6 4 180 size Angle

The angle sum of the exterior angles of a n-sided polygon is 360o.

Find the size of each interior angle of a regular decagon.

Sum of exterior angles = 360o Exterior angles = o 10 360       = 36o

Interior angles = 144o (angle sum of straight angle equals 180o)

Similar Triangles

Two triangles are similar if two angles of one triangle are equal to two angles of the other triangle.

Prove that ABC and DCA are similar.

 

*

*

In ABC and DCA

ABC = ACD (given)

BAC = ADC (given)

ABC ||| DCA (equiangular) *

* The abbreviations AA or AAA are not to be accepted

Two triangles are similar if the ratio of two pairs sides are equal and the angles included by these sides are equal.

Prove that ABC and ACD are similar.

36 16 24 * * A B C D

In ABC and ACD

BCA = ACD (given)

2 3 24 36   AC BC 2 3 16 24_  DC AC

BCA ||| ACD (sides about equal angles are in

the same ratio) * * sides about equal angles are in proportion

Two triangles are similar if the ratio of the three pairs of sides are equal.

Prove that ABC and ACD are similar.

A B C D 12 16 24 18 32

In ABC and ACD

3 4 12 16   CD AB 3 4 24 32 _  AC BC 3 4 18 24   AD AC

ABC ||| DCA (three pairs of sides in the same

ratio) * * three pairs of sides in proportion

Example:

Given that AB //PQ, find the value of x.

9 cm Q P C B A x cm 12 cm 8 cm In ABC and PQC PQC ABC

 (corresponding angles are equal

as AB //PQ) PCQ ACB  (common) PQC ABC    ||| (equiangular) 12 20 9  x

(corresponding sides in similar triangles are in the same ratio) *

12 20 9  x 15  x

* corresponding sides in similar triangles are in proportion

Congruent Triangles

Two triangles are congruent if three sides of one triangle are equal to three sides of the other triangle.

Given that AC = BD and AB = CD. Prove that CAB BDC.

12 8 12 8 A B C D In CAB and BDC.

AC = BD (both 8) or (given) or (data) AB = CD (both 12) or (given) or (data) CB = CB (common) or CB is common CAB BDC (SSS) or In CAB and BDC. AC = BD = 8 AB = CD = 12 CB = CB (common) or CB is common CAB BDC (SSS)

Two triangles are congruent if two sides of one triangle are equal to two sides of the other triangle and the angles included by these sides are equal.

Given that AC = BD and CAB = DBA.

Prove that CAB DBA.

=

=

A B

C

D

In CAB and DBA

AB = AB (common) or AB is common AC = BD (given)

CAB = DBA (given)

CAB DBA (SAS)

Two triangles are congruent if two angles of one triangle are equal to two angles of the other triangle and one pair of corresponding sides are equal.

Given that AB = CD and EAB = ECD.

Prove that ABE CDE.

=

=

*

*

In ABE and CDE. AB = CD (given)

EAB = ECD (given)

AEB = CED (vertically opposite angles are

equal)

Two right- angled triangles are congruent if their hypotenuses are equal and a pair of sides are also equal.

Given that CD = AD. Prove that ABD CBD.

=

=

Intercepts and Parallels

An interval joining the midpoints of the sides of a triangle is parallel to the third side and half its length.

E and F are midpoints of AB and AC. G and H are midpoints of FB and FC.

Prove that EF = GH.

B C

A

E F

G H

EF=½BC (interval joining midpoints of sides of ABC

 is half the length 3rd side) Similarly in BFC , GH=½BC

 EF = GH

(Note: It can also be proven that EF and GH are parallel)

An interval parallel to a side of a triangle divides the other sides in the same ratio. (converse is true)

Find the value of x.

>

>

(interval parallel to side of ABC divides other sides in same ratio)

x = 12

Parallel lines preserve the ratio of intercepts on transversals. (converse is not true)

Find the value of x.

>

>

>

(parallel lines preserve the ratios of intercepts on transversals) *

x = 24

* intercepts on parallel lines are in the same ratio * intercepts on parallel lines are in proportion

Pythagoras’ Theorem

Pythagoras’ Theorem: The square on the hypotenuse equals the sum of the squares on the other two sides in a right angled triangle.

Find the value of x.

12 15 x 2 2 2 15 12   x (Pythagoras’ Theorem) 9 81 144 225 2     x x or 9  x (3,4,5 Pythagorean Triad)

A triangle is right-angled if the square on the hypotenuse equals the sum of the squares on the other two sides (converse of Pythagoras’ Theorem)

Prove that ABC is right-angled

8 cm 10 cm 6 cm A C B 2 2 2 2 2 2 2 2 2 100 64 36 8 6 100 10 BC AC AB AC AB BC           

ABC is right-angled (Converse of Pythagoras’

Quadrilateral Properties

Trapezium

One pair of sides of a trapezium are parallel

The non-parallel sides of an isosceles trapezium are equal

Parallelogram

The opposite sides of a parallelogram are parallel The opposite sides of a parallelogram are equal The opposite angles of a parallelogram are equal The diagonals of a parallelogram bisect each other Adjacent angles are supplementary

A parallelogram has point symmetry

Kite

Two pairs of adjacent sides of a kite are equal One diagonal of a kite bisects the other diagonal One diagonal of a kite bisects the opposite angles The diagonals of a kite are perpendicular

A kite has one axis of symmetry

Rhombus

The opposite sides of a rhombus are parallel All sides of a rhombus are equal

The opposite angles of a rhombus are equal

The diagonals of a rhombus bisect the opposite angles The diagonals of a rhombus bisect each other

The diagonals of a rhombus are perpendicular A rhombus has two axes of symmetry

A rhombus has point symmetry

Rectangle

The opposite sides of a rectangle are parallel The opposite sides of a rectangle are equal All angles at the vertices of a rectangle are 90o The diagonals of a rectangle are equal

The diagonals of a rectangle bisect each other A rectangle has two axes of symmetry

A rectangle has point symmetry

Square

Opposite sides of a square are parallel All sides of a square are equal

All angles at the vertices of a square are 90o The diagonals of a square are equal

The diagonals of a square bisect the opposite angles The diagonals of a square bisect each other

The diagonals of a square are perpendicular A square has four axes of symmetry

Sufficiency conditions for Quadrilaterals

A quadrilateral is a parallelogram if

 both pairs of opposite sides are parallel or

 both pairs of opposite sides are equal or

 both pairs of opposite angles are equal or

 the diagonals bisect each other or

 one pair of sides are equal and parallel

Sufficiency conditions for rhombuses

A quadrilateral is a rhombus if

 all sides are equal or

 the diagonals bisect each other at right angles or

 the diagonals bisect each vertex angle

Sufficiency conditions for rectangles

A quadrilateral is a rectangle if

 all four angles are equal or

 the diagonals are equal and bisect each other

Sufficiency condition for squares

A quadrilateral is a square if

 the diagonals are equal and bisect each other at right angles

Sufficiency condition for kites

A quadrilateral is a kite if

 the diagonals are perpendicular and one is bisected by the other

Circles and Chords or Arcs

Equal chords subtend equal arcs on a circle. (converse is true) Equal arcs subtend equal chords on a circle. (converse is true)

Equal chords subtend equal angles at the centre of a circle. (converse is true)

AB = EF. Find the value of x.

x 68

O

E

A

F

B

x = 68 (equal chords subtend equal angles at the

centre)

Equal arcs subtend equal angles at the centre of a circle. (converse is true)

arc AB = arc EF. Find the value of x.

x 68

O

E

A

F

B

x = 68 (equal arcs subtend equal angles at the

centre)

Equal angles at the centre of a circle subtend equal chords. (converse is true)

Chord EF = 16cm, find the length of chord AB.

O

F

E

B

A

75



75



AB = 16 cm (equal angles at the centre subtend

Equal angles at the centre of a circle subtend equal arcs. (converse is true)

arc EF = 16cm, find the length of arc AB.

16 cm

O

F

E

B

A

75



75



arc AB = 16 cm (equal angles at the centre subtend

equal arcs)

A line through the centre of a circle perpendicular to a chord bisects the chord. (converse is true)

O is the centre of the circle. Find the length of AP.

8 cm

O B

A P

AP = 8 cm (interval through center perpendicular to

chord AB bisects the chord)

A line through the centre of a circle that bisects a chord is perpendicular to the chord. (converse is true)

Find the size of OEB.

6 cm 6 cm

E

O

B

C

Chords equidistant from the centre of a circle are equal. (converse is true)

Find the length of XY.

5cm

=

=

AB = 10 cm (interval through centre perpendicular

to chord AB bisects the chord)

XY = 10 cm (chords equidistant from the centre of

a circle are equal)

Equal chords are equidistant from the centre of a circle. (converse is true)

Find the length of OL.

7 7 5 7 7

L

M

O

H

I

G

F

OL = 10 (equal chords are the equidistant from the

Angles in Circles

The angle at the centre of a circle is twice the angle at the circumference standing on the same arc. The angle at the circumference of a circle is half the angle at the centre standing on the same arc.

(i) Find the value of y.

O B A C 54 y

(ii) Find the value of x.

O B A C 94 x

(i) y = 108 (angle at centre equals twice angle circumference standing on arc AB)

Note: use arc AB and not chord AB – the statement is not necessarily true for chords

(ii) x = 47 (angle at circumference equals half angle at centre standing on arc AB)

Angles at the circumference standing on the same arc are equal or

Angles at the circumference in the same segment are equal. (converse is true)

Find the value of x.

S O P R Q 41 x

x = 41 (angles at the circumference on the same

arc PQ are equal)

(Note: use arc PQ and not chord PQ – the statement is not necessarily true for chords)

or

x = 41 (angles at the circumference in the same

Equal arcs subtend equal angles at the circumference. (converse is true)

arc AB = arc CD. Find the value of x.

x 37

D

A

B

C

F

E

x = 37 (Equal arcs subtend equal angles at the

circumference)

Note: the statement is not necessarily true for equal chords

Equal angles at the circumference subtend equal arcs.

Find the length of arc PQ.

8 cm 25 25 N Q Y X P M

PQ = 8 cm (Equal angles at the circumference

subtend equal arcs)

The angle at the circumference in a semi-circle is 90o.

AB is a diameter. Find the value of x.

38 x

A

O

B

P

A (angle at circumference in semi-circle

equals 90o)

x + 128 = 180 (angle sum of APB equals 180o)

x = 52

A right angle at the circumference subtends a diameter

IfACˆB90then AB is a diameter.

A radius (diameter) of a circle is perpendicular to the tangent at their point of contact

STU is a tangent at T. Find the size of TOU.

26

O

T

U

S

OTU = 90o (radius is perpendicular to tangent at point of contact)

TOU + 116o = 180o (angle sum of OUT equals

180o)

TOU = 64o

The angle between a tangent and a chord equals the angle at the circumference in the alternate segment.

Find the size of RTN.

93 T N M R S

RTN = 93o (angle between tangent and chord equals angle at circumference in alternate segment)

Cyclic Quadrilaterals

The opposite angles of a cyclic quadrilateral are supplementary. (converse is true)

Find the value of x.

C D B A 87 xo o

x + 87 = 180 (opposite angles of cyclic

quadrilateral ABCD are supplementary)

x = 93

* opposite angles of a cyclic quadrilateral are supplementary

The exterior angle of a cyclic quadrilateral equals the opposite (or remote) interior angle. (converse is true)

Find the size of ADE.

D C E B A o 112

ADE = 112o (exterior angle of cyclic quadrilateral ABCD equals opposite interior angle)

or

ADE = 112o (exterior angle of cyclic quadrilateral ABCD equals remote interior angle)

Intercept Theorems

The product of the intercepts on intersecting chords are equal. (converse is true)

Find the value of x.

x 12 Q A P B 18 8 x  8 = 12 18 (product of intercepts on

intersecting chords are equal)

x = 27

The product of the intercepts on intersecting secants are equal.

Find the value of x.

x A P Q B T 9 3 12





9x = 99

x = 11

The square of the intercept on tangent to a circle equals the product of the intercepts on the secant.

Find the value of x.

x 12 T B A P 4 4 16 2  

x (square of intercept on tangent to circle equals product of intercepts on secant)

x2 = 64

Intercepts on tangents drawn from a point to a circle are equal.

Find the value of x.

x

35

x = 35 (intercepts on tangents

from a point to a circle are equal)

Converses of Cyclic Quadrilateral theorems

If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

XA and YB are altitudes of XYZ. Prove that AZBP

is a cyclic quadrilateral. X Y Z A B P

YBZ = 90o (YB is an altitude)

XAZ = 90o (XA is an altitude)

PBZ + PAZ = 180o

AZBP is cyclic (opposite angles are

supplementary)

If the exterior angle of a quadrilateral equals the opposite interior angle then the quadrilateral is cyclic.

Prove that ABCD is a cyclic quadrilateral.

87 87 A B C D T o o DAB = TCB (both 87o)

ABCD is a cyclic (exterior angle equals opposite

interior angle)

If a side of a quadrilateral subtends equal angles at the other two vertices then the quadrilateral is cyclic.

OR

If an interval subtends equal angles on the same side at two points then the ends of the interval and the two points are concyclic.

XA and YB are altitudes of XYZ. Prove that XBAY

are the vertices of a cyclic quadrilateral.

X Y Z A B P

XBY = 90o (YB is an altitude)

XAY = 90o (XA is an altitude)

XBA = XAY = 90o

 XBAY is cyclic (XY subtends equal angles on

If the product of the intercepts on intersecting intervals are equal then the endpoints of the intervals are concyclic.

Prove that points A, C, B and D are concyclic.

A B C D F 4 6 9 6 36    FB DF FC AF

 A, C, B and D are concyclic (product of