Double Angle Triangles inside Circles Angles connected by a chord Tangents to a circle Cyclic Quadrilaterals 2x x This is the ARC o Centre of Circle The Angle x subtended at the centre of a circle by an arc is twice the size of the angle on the circumference subtended by the same ar ID: 421822
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Slide1
Circle theorems
Double Angle
Triangles inside Circles
Angles connected by a
chord
Tangents to a circle
Cyclic QuadrilateralsSlide2
2x
x
This is the ARC
o
Centre of Circle
The Angle
x
subtended at the centre of a circle by an arc is twice the size of the angle on the circumference subtended by the same arc.Slide3
2x
x
o
This is the ARC
Centre of Circle
Angle subtended at the Centre is twice the angle at the circumference
Case 2Slide4
x
x
x
We are ALL EQUAL
This is the Arc
Angles Subtended in the same segment
of a circle are equal
Chop Sticks
Minor Segment
Major SegmentSlide5
o
A
B
C
D
x
180-
x
If this angle was 60
0
then angle
BCD would be 180
0
-60
0
=120
0
120
0
Cyclic Quadrilateral
Points which lie on the circumference of the
same circle are called cyclic (or concyclic)
points. A
cyclic
quadrilateral is a quadrilateral
with all its four corners (vertices) on the
circumference of the same circle.Slide6
Tangents
T
A
B
O
TA=TB
NB Triangles
OBT and OAT
are CONGRUENT!
Tangent
TangentSlide7
Major Segment
Minor Segment
A
B
C
E
D
The Shaded Segment BED
is called the
alternate segment
to the angle CBD
The angle between a tangent to a circle and a chord drawn through the point
of contact is equal to any angle subtended by the chord at the circumference in
the alternate segment Slide8
Centre of Circle
Diameter
The angle in a semi circle is 90 degrees!Slide9
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The angle at the centreSlide11
25°
x
160°
100°
60°
135°
90°
x
x
x
x
x
1
2
3
6
4
5
Answers
1) 50
2)120
3)180
4)50
5)67.5
6)80
Home
Double angle theoremSlide12
Right angles in a semicircleSlide13
60°
x
1
23
31
2
72°
x
x
x
x
y
y
x
100°
x
30°
22°
y
Answers
1) X=30
2)x=18
3)x=45
4)X=40 y=40
5)x=30 y= 120
6)x=22 y=136
x
Home
Triangles inside circlesSlide14
Angles in the same segmentSlide15
25°
x
1
2
3
6
45
y
15°
y
z
z
x
y
x
z
x
y
y
z
x
25°
53°
30°
z
y
x
80°
17°
95°
35°
40°
125°
15°
40°
10°
100°
Answers
1) x=25 y=15
2)x=125 y= 40 z=15
3)x=10 y=70 z=100
4)X=105 y=40 z=35
5)x=53 y= 30 z=72
6)x=85 y=80 z=17
Home
Angles connected by a chord
(off the same arc)Slide16
The tangent and the radiusSlide17
Two tangents from a pointSlide18
40°
x
y
z
3
120°
x
4
1
140°
x
2
x
35
°
1
y
z
Home
Tangents to a circle
Answers:
x=55
x=40
x=50 y=50 z=40
x=60 y=60 z=30Slide19
Angles in a cyclic quadrilateralSlide20
x
y
x
y
x
y
95°
110°
54°
75°
20°
80°
x
2a
4b
15°
70°
a
b
1
25°
y
z
w
2
3
4
5
Answers
1) x=70 y=85
2)x=126 y=105
3)x=100 y=160
4)w=15 x=70 y=65 z= 25
5)a=60 b=36
Home
Cyclic QuadrilateralsSlide21
The alternate segment theorem