Geometry Theorems and Proofs

8/2/2019 Geometry Theorems and Proofs

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JRAHS Geometry Proofs (SV) 30/5/05 1

GEOMETRY THEOREMS AND PROOFS

Rational:The policy of the JRAHS 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 ofx.

C

B

Ax

42

73

EQUATION REASON COMMENT

65180115

1807342

xx

x

(Angle sum of

180equalsABC )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) abbreviation such as coint, alt, vert opp, etc are not to be used words are to be written infull

(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, PQST(4) If the geometric shape is not labelled then the students may introduce their own labels or refer to theshape in general terms such as angle sum of triangle = 180o or angle sum of straight angle = 180o


2/27


Revolution, Straight Angles, Adjacent angles, Vertically opposite angles

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

Find the value ofx.

P

165

60

x2x

D

C

B

A

360165602 xx (angle sum at a pointPequals 360

o)

3602253 x 1353 x

45x

A right angle equals 90o.

AB is perpendicular toBC. Find the value ofx.

D

CB

A

x36

9036 x (angle sum of right angle ABCequals 90o)

54x

A straight angle equals 180o.

FMJis a straight segment. Find the value ofx.

J

I

H

G

F M

50

46 4x

2x

180504642 xx (angle sum of straightangleFMJequals 180o)

180966 x 846 x

14x

Three points are collinear if they form a straight angle

Given thatAKB is a straight line.Prove that the pointsP,Kand Q are collinear.

Q

P

K

B

A

72

3x

2x

18023 xx (angle sum of straight angleAKBequals 180o)

1805 x36x

180

72363

723 xQKP

P,Kand Q are collinear (PKQ is a straightangle) *

* PKQ equals 180o


3/27


Vertically opposite angles are equal.

ACandDEare straight lines. Find the value ofy.

y

29

D

B

E C

A

67

6729 y (vertically opposite angles are equal)

38y


4/27


Angles and Parallel LinesAlternate angles on parallel lines are equal.

All lines are straight. Find the value ofx.

>>

>>

A B

C D

E

H

F

G

x

59o

o

59x (alternate angles are equal asAB||CD)

Corresponding angles on parallel lines are equal.


>>

>>

A

B

C D

E

F

G

H

137

xo

o

137x (corresponding angles are equal asAB||CD)

Cointerior angles on parallel lines are supplementary.


>>

>>

A

B

C D

E

F

G

H

125

x o

o

180125 x (cointerior angles aresupplementary asAB||CD)

55x


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

Prove thatAB // CD

73

73H

G

C D

A

B

E

F

GHDAGH (both 73o) **CDAB || (alternate angles are equal)

** equality of the angles involved must be clearlyindicated

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

Prove thatAB // CD

65

65H

G

C D

A

B

E

F

EGB = GHD (both 65o) **

CDAB || (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 thatPR //KM

56

124

L

Q

K M

P

R

X

Y

RQL + QLM= 124o + 56o **= 180o

KMPR || (cointerior angles are

supplementary)

* RQL + QLR = 180o

** supplementary nature of the angles involved

must be clearly indicated


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Angles in PolygonsThe angle sum of a triangle is 180

o.

Find the value ofx.

A

B

C

x34

67

o

o

o

1803467 x (angle sum of ABC equals180

o)

180101 x 79x

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

Find the value ofx.

A DC

B

x

47

68oo

o

4768 x (exterior angle of ABC equals sumof the two opposite interior angles)

115x

* exterior angle of ABC equals sum of remoteinterior angles

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

Find the value ofx.

A C

B

x

157

128o

o

o

360128157 x (sum of exterior angles ofABC equals 360o)

360285 x 75x

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

Find the value ofx.

||

=

A B

C

54

xo

o

54x (equal angles are opposite equal sides inABC ) *

* base angles of isosceles ABC are equal


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

Find the value ofx.

12

15

x

A

B C

65o

65o

15x (equal sides are opposite equal angles inABC )

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

ABC is equilateral.ECandDB are anglebisectors and meet atP. Find the size ofCPB.

BC

A

PD E

ACB = 60o

(all angles of an equilateral triangleare 60

o)

similarly ABC= 60o

ECB = 30o (ECbisects ACB)similarly DBC= 30oCPB + 60o = 180o (angle sum of PCB equals

180o)

CPB = 120o

The angle sum of a quadrilateral is 360o.

Find the value ofx.

A

B

C

D

o

o

o

ox

3x

130

70

3602004 x (angle sum of quadrilateralABCDequals 360o)

40

1604

x

x


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The angle sum of a n-sided polygon is 180(n 2)o

or (2n 4) right angles.

Find the value ofx.

106

87

x

165

92

B

C

D

E

A

Angle sum of a pentagon = 3 180o= 540

o

x + 450 = 540 (angle sum of pentagon equals

540o)

x = 90

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

o

2180

n

n

.

Find the size of each interior angle of a regular

hexagon

120

6

4180sizeAngle

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

10360

= 36o

Interior angles = 144o (angle sum of straight angle

equals 180o)


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

Prove that ABCand DCA are similar.

**

A

B

C

D

In ABCand DCAABC= ACD (given)BAC= ADC(given)ABC||| DCA (equiangular) *

* The abbreviationsAA orAAA are not to beaccepted

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

sides are equal.

Prove that ABCand ACD are similar.

36

16

24

**

A

B

C

D

In ABCand ACDBCA = ACD (given)

2

3

24

36

AC

BC

2

3

16

24

DC

AC

BCA ||| ACD (sides about equal angles are inthe 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 ABCand ACD are similar.

A

B C

D

12

16

24

18

32

In ABCand ACD

3

4

12

16

CD

AB

34

2432

ACBC

3

4

18

24

AD

AC

ABC||| DCA (three pairs of sides in the sameratio) *

* three pairs of sides in proportion


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Example problem:

Given that PQAB // , find the value of x.

9 cm

Q

P

CB

A

x cm

12 cm8 cm

In ABC and PQC

PQCABC (corresponding angles are equal

as PQAB // )

PCQACB (common)

PQCABC ||| (equiangular)

1220

9 x (corresponding sides in similar triangles

are in the same ratio) *

12

209x

15x

* corresponding sides in similar triangles are in

proportion


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Congruent TrianglesTwo triangles are congruent if three sides of one triangle are equal to three sides of the other

triangle.

Given thatAC=BD andAB = CD.

Prove that CABBDC.

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) orCB is common

CABBDC(SSS)

or

In CAB and BDC.AC=BD = 8

AB = CD = 12

CB = CB (common) orCB is common

CABBDC(SSS)

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

Given thatAC=BD and CAB = DBA.Prove that CABDBA.

= =

A B

CD

In CAB and DBAAB =AB (common) orAB is common

AC=BD (given)

CAB = DBA (given)CABDBA (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 thatAB = CD and EAB = ECD.Prove that ABECDE.

= =

A

B

C

D

E

* *

In ABEand CDE.AB = CD (given)

EAB = ECD (given)AEB = CED (vertically opposite angles are

equal)

ABECDE(AAS)


12/27


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

equal.

Given that CD =AD. Prove that ABDCBD.

=

=

A

BD

C

In ABD and CBDBCD = BAD (both 90o)CD =AD (given)

DB =DB (common)

ABDCBD (RHS)


13/27


Intercepts and ParallelsAn interval joining the midpoints of the sides of a triangle is parallel to the third side and half its

length.

EandFare midpoints ofAB andAC.

G andHare midpoints ofFB andFC.

Prove thatEF= GH.

B C

A

E F

G H

EF=BC(interval joining midpoints of sides of

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

EF= GH

(Note: It can also be proven thatEFand GHare

parallel)

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

Find the value ofx.

>

>

B C

A

I J

x

15 9

20

15

20

9

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 ofx.

>

>

>

x

24

32

18

24

18

32

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


14/27


Circles and Chords or ArcsEqual 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 ofx.

x68

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 = arcEF. Find the value ofx.

x68O

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)

ChordEF= 16cm, find the length of chordAB.

O

F

E

B

A

7575

AB = 16 cm (equal angles at the centre subtend

equal chords)


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

arcEF= 16cm, find the length of arcAB.

16 cm

O

F

E

B

A

7575

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 ofAP.

8 cm

O

B

A

P

AP= 8 cm (interval through center perpendicular tochordAB 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 ofOEB.

6 cm

6 cm

E

O

B

C

chordthelar toperpendicuischord

bisectingcentrethroughinterval90

BCOEB

NOTE: It can be proven that the perpendicular bisector of a chord passes through the center of the

circle.


16/27


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

Find the length ofXY.

5cm

==

O

B

AP

Y

XQ

AB = 10 cm (interval through centre perpendicular

to chordAB 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 ofOL.

7

75

7

7

LM

OH

I

G

F

IH=FG = 14

OL = 10 (equal chords are the equidistant from the

centre)


17/27


Angles in CirclesThe 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 ofx.

O

B

A

C

94

x

(i)y = 108 (angle at centre equals twice angle

circumference standing on arcAB)

Note: use arcAB and not chordAB thestatement is not necessarily true for

chords

(ii)x = 47 (angle at circumference equals half

angle at centre standing on arcAB)

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 ofx.

S

O

P

R

Q

41

x

x = 41 (angles at the circumference on the same

arcPQ are equal)

(Note: use arcPQ and not chordPQ the

statement is not necessarily true for chords)

or

x = 41 (angles at the circumference in the same

segment equal)


18/27


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

arcAB = arc CD. Find the value ofx.

x

37

D

A

B

C

FE

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 arcPQ.

8 cm

25

25N

Q

Y

X

P

M

PQ = 8 cm (Equal angles at the circumferencesubtend equal arcs)

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

AB is a diameter. Find the value ofx.

38x

A O B

P

90BPA (angle at circumference in semi-circleequals 90o)

x + 128 = 180 (angle sum ofAPB equals 180o)x = 52

A right angle at the circumference subtends a diameter

If 90BCA thenAB is a diameter.

BA

C


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

STUis a tangent at T. Find the size of TOU.

26

O

T

U

S

OTU= 90o (radius is perpendicular to tangent atpoint of contact)

TOU+ 116o = 180o (angle sum ofOUTequals180o)

TOU= 64o

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

segment.

Find the size ofRTN.

93

T

N

M

R

S

RTN= 93o (angle between tangent and chordequals angle at circumference in

alternate segment)


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

Find the value ofx.

C

D

B

A

87

xo

o

x + 87 = 180 (opposite angles of cyclic

quadrilateralABCD aresupplementary)

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 ofADE.

DCE

B

A

o

112

ADE= 112o (exterior angle of cyclicquadrilateralABCD equals

opposite interior angle)

or

ADE= 112o

(exterior angle of cyclicquadrilateralABCD equalsremote interior angle)


21/27


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

Find the value ofx.

x

12

QA

P

B

18

8

x 8 = 12 18 (product of intercepts onintersecting chords are equal)

x = 27

The product of the intercepts on intersecting secants are equal.

Find the value ofx.

x

A

P Q

B

T

93

12

121599 x (product of intercepts onintersecting secants are equal)

9x + 81 = 180

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 ofx.

x

12

T

B

A

P4

4162 x (square of intercept on tangent tocircle equals product of intercepts

on secant)

x2 = 64

x = 8 (length > 0)


22/27


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

Find the value ofx.

x

35

x = 35 (intercepts on tangentsfrom a point to a circle

are equal)

The line joining the centers of two circles passes through their point of contact


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Converses of Cyclic Quadrilateral theoremsIf the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

XA and YB are altitudes ofXYZ. Prove thatAZBPis a cyclic quadrilateral.

X

Y

Z

A

B

P

YBZ= 90o (YB is an altitude)XAZ= 90o (XA is an altitude)PBZ+ PAZ= 180o

AZBPis cyclic (opposite angles are

supplementary)

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

cyclic.

Prove thatABCD is a cyclic quadrilateral.

87

87

AB

C

D

T

o

o

DAB = TCB (both 87o)

ABCD is a cyclic (exterior angle equals oppositeinterior 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 ofXYZ. Prove thatXBAY

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 XBAYis cyclic (XYsubtends equal angles on

the same side atA andB)


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

are concyclic.

Prove that pointsA, C,B andD are concyclic.

A

B

C

D

F

4

69

6

36 FCDFFBAF

A, C,B andD are concyclic (product ofintercepts are equal)


25/27


Pythagoras TheoremPythagoras 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 ofx.

12

15x

222 1512 x (Pythagoras Theorem)

9

81

1442252

x

x

or

9x (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 thatABCis right-angled

8 cm

10 cm

6 cm

A C

B

222

2222

22

100

6436

86

10010

BCACAB

ACAB

BC

ABCis right-angled (Pythagoras theoremconverse)


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Quadrilateral PropertiesTrapezium

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

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 parallelAll 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

A square has point symmetry


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Sufficiency conditions for QuadrilateralsSufficiency conditions for parallelograms

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

Geometry Theorems and Proofs

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Transcript of Geometry Theorems and Proofs