Transcript ppt 2

Specialist Maths
Geometry Proofs
Week 2
Parallel Lines
Corresponding Angle
a
Alternate Angles
a
a
a
Allied or Co-interior Angles
a + b = 1800
a
b
Triangles
b
a
c
a
b
Exterior angle of a triangle
c=a+b
c
a + b + c = 180o
b b
a
Isosceles Triangle
a
Quadrilaterals
a
b
c
b
a
d
b
a
a + b + c + d = 360o
b b
a
a
a
b
b
a
Congruent triangles
SSS
SAS
AA cor S
b
a
a
b
a
a
RHS
Similar Triangles
To prove triangles are similar,
show two set of angles are equal,
then the third set must also be equal.


a

b
kb
ka

c


kc
Circle Theorems
Angle in a semi circle
Tangents from
external point
Chord of a circle
Angle at centre
a
Tangent-radius
Angles subtended
from same arc
a
2a
a
Intersecting Chords
AX.BX = CX.DX
AX.BX =(PX)2
D
A
A
X
B
P
C
X
B
A
C
D
AX.BX = CX.DX
B
X
More Theorems
Angle between tangent & chord
Mid Point Theorem
M
N
a
A
MN is parallel to
AB and half its length
Angles in Cyclic
Quadrilateral
a
B
a
a+ b = 180o
b
Tests for Cyclic quadrilaterals or
Concyclic Points
Test 1
Test 2
b
a
b
a
Show a + b = 180o
Show a = b
Example 6 Ex 5B1)
Show that tria ngles ABE and ADC are similar.
Hence find the value of x.
4 cm
E
α
D
2 cm
α
A
x cm
B
4 cm
C
Solution 6
Show that tria ngles ABE and ADC are similar.
Hence find the value of x.
4 cm
E
α
D
2 cm
α
A
x cm
B
4 cm
C
Example 7 (Ex 5B1)
Find a relationdh ip between th e unknown angles :
P

O
(PQ is tangent)
α
β
R
Q
Solution 7
Find a relationdh ip between th e unknown angles :
(PQ is tangent)
P

O
α
β
R
Q
Example 8 (Ex 5B1)
(1) Prove MQX is congruent to MRY.
(2) Prove PQR is isosceles.
P
Y
X
Q
M
R
Solution 8
(1) Prove MQX is congruent to MRY.
(2) Prove PQR is isosceles.
P
Y
X
Q
M
R
Example 9 (Ex 5B2)
ABCD is a cyclic quadrilate ral. BE bisects ABC
and DF bisects ADC. Prove that EF is the diameter
of the circle.
C
D
B
β
α
F
A
E
Solution 9
C
α
B
β α
β
D
β
F
A
E
Example 10 (Ex 5B2)
A, B & C are three points on a circle. D and E lie on AB
and AC respective ly such that DE is parallel to BC. If
CD produced and BE produced meet the circle at X and
Y respective ly, show that XYED is a cyclic quadrilate ral.
Solution 10
A, B & C are three points on a circle. D and E lie on AB
and AC respective ly such that DE is parallel to BC. If
CD produced and BE produced meet the circle at X and
Y respective ly, show that XYED is a cyclic quadrilate ral.
Example 11(Ex 5B2)
XYZ is an isosceles triangle where XY  XZ. A circle
is drawn thro ugh X and Y cutting XZ at A and B. Show
that trian gle ABZ is isosceles
X
A
Y
Z
B
Solution 11
XYZ is an isosceles triangle where XY  XZ. A circle
is drawn thro ugh X and Y cutting XZ at A and B. Show
that trian gle ABZ is isosceles
This Week
•
•
•
•
Test book pages 184 to 202
Exercise 5B1 questions 1-14
Exercise 5B2 questions 1-20
Review Sets 5A – 5C