Specialist Maths

Geometry Proofs

Week 2

Parallel LinesCorresponding Angle Alternate Angles

Allied or Co-interior Angles

a

a

a

a

a

ba + b = 1800

Triangles

a

b c

a + b + c = 180o

a

b

c

Exterior angle of a trianglec = a + b

a a

b b Isosceles Triangle

Quadrilaterals

a b

c d

a + b + c + d = 360oa

ab

b

aa

aab b

bb

Congruent triangles

SSS SAS AA cor S RHS

a

a a

a

b

b

Similar Triangles

a

c

b

kc

kbka

To prove triangles are similar,show two set of angles are equal,then the third set must also be equal.

Circle Theorems

a

2a a

a

Angle in a semi circle Chord of a circle Tangent-radius

Tangents fromexternal point

Angle at centre Angles subtendedfrom same arc

Intersecting Chords

A

A

A

B

B B

C

C

D

D

X

X

X

P

AX.BX =(PX)2

AX.BX = CX.DX

AX.BX = CX.DX

More Theorems

AB

M N

Mid Point Theorem

MN is parallel to AB and half its length

a

a

Angle between tangent & chord

b

a

a+ b = 180o

Angles in CyclicQuadrilateral

Tests for Cyclic quadrilaterals or Concyclic Points

a

b

b

a

Test 1

Show a + b = 180o

Show a = b

Test 2

Example 6 Ex 5B1)

. of value thefind Hence

similar. are ADC and ABE ngles that triaShow

xD

CBA

E

α

α

x cm 4 cm

4 cm

2 cm

Solution 6

. of value thefind Hence

similar. are ADC and ABE ngles that triaShow

x

CBA

E

α

α

x cm 4 cm

4 cm

2 cm

D

Example 7 (Ex 5B1)

:anglesunknown ebetween th iprelationdh a Find

O

P

Q

R

α β

(PQ is tangent)

Solution 7:anglesunknown ebetween th iprelationdh a Find

(PQ is tangent)

O

P

Q

R

α β

Example 8 (Ex 5B1)

isosceles. is PQR Prove (2)

MRY. tocongruent is MQX Prove (1)

P

Y

RMQ

X

Solution 8

isosceles. is PQR Prove (2)

MRY. tocongruent is MQX Prove (1)

Y

RMQ

X

P

Example 9 (Ex 5B2)

circle. theof

diameter theis EF that Prove ADC. bisects DF and

ABC bisects BE ral.quadrilate cyclic a is ABCD

A

B

C

D

E

F

αβ

Solution 9

A

B

C

D

E

F

αβα

β

β

Example 10 (Ex 5B2)

ral.quadrilate cyclic a is XYED that show ly,respective Y

and Xat circle meet the produced BE and produced CD

If BC. toparallel is DEsuch that ly respective AC and

ABon lie E and D circle. aon points threeare C & B A,

Solution 10

ral.quadrilate cyclic a is XYED that show ly,respective Y

and Xat circle meet the produced BE and produced CD

If BC. toparallel is DEsuch that ly respective AC and

ABon lie E and D circle. aon points threeare C & B A,

Example 11(Ex 5B2)

isosceles is ABZ glethat trian

Show B. andA at XZ cutting Y and Xugh drawn thro is

circleA XZ. XY where triangleisoscelesan is XYZ

X

Y Z

A

B

Solution 11

isosceles is ABZ glethat trian

Show B. andA at XZ cutting Y and Xugh drawn thro is

circleA XZ. XY where triangleisoscelesan is XYZ

This Week

• Test book pages 184 to 202

• Exercise 5B1 questions 1-14

• Exercise 5B2 questions 1-20

• Review Sets 5A – 5C