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Shape and Space
CIRCLE GEOMETRY
Circle Geometry
Rule 1 : ANGLE IN A SEMICIRCLE = 90°
A triangle drawn from the two ends of a diameter will always make an angle of 90° where it hits the edge of the circle, no matter where it hits.
40°
a
a = 50°
EXERCISES
What is the size of each angle? Copy each diagram.
65°
a 45°
c
63°
d81°
b
Rule 2 : A TANGENT AND RADIUS MEET AT 90°
A tangent is a line that just touches the edge of a curve. If a tangent and radius meet at the same point, then the angle they make is exactly 90°.
a = 30°
a
60°
Circle Geometry
EXERCISES
What is the size of each angle? Copy each diagram.
47°a
33°
b
59°c
71°
d
Rule 3 : TWO RADII CREATE AN ISOSCELES TRIANGLE
These triangles are constructed using two radii. Therefore they are isosceles triangles.
a = 30°
a 120°
Circle Geometry
EXERCISES
What is the size of each angle? Copy each diagram.
67°a
33°c
110°
b
220°
d
Circle Geometry
Rule 4 : THE PERPENDICULAR CORD BISECTOR IS A DIAMETER
No matter where you draw the cord, the line that cuts it exactly in half at 90° will go through the center of the circle – the diameter line.
a = 36°
a
54°
EXERCISES
What is the size of each angle? Copy each diagram.
a
87°
b
67°
c
41°
d
77°
Circle Geometry
Rule 5 : ANGLES IN THE SAME SEGMENT ARE EQUAL
All triangles drawn from a chord will have the same angle where they touch the circle. Also the two angles on opposite sides of the chord add up to 180°
a = 50°b = 130°
a
a
bb
a + b = 180°50°
a
b
EXERCISES
What is the size of each angle? Copy each diagram.
111° g
h
43°
a
b
31°
e
f
d
76°c
Circle Geometry
Rule 6 : ANGLE AT THE CENTRE IS TWICE THE ANGLE AT THE EDGE
Within the same segment, the angle made at the centre of a circle is exactly double the angle formed by the same chord at the edge of the circle.
a = 126°
a
a
a 2a
63°
a
EXERCISES
What is the size of each angle? Copy each diagram.
79°
d
a
122°
72°
b
128°
c
Circle Geometry
°
Rule 7 : OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL ADDS TO 180°
A cyclic quadrilateral is a four sided shape with every corner touching the circle. Both pairs of opposite angles add up to 180°
a = 93°b = 101°
a b
d
c
a + c = 180°b + d = 180°
a
b87°
79°
EXERCISES
What is the size of each angle? Copy each diagram.
a
b111°
89°
e
f
101°
81°
cd
72°
61°g
h
132°
59°
Circle Geometry
Rule 8 : EQALITY OF TANGENTS FROM A POINT
The two tangents drawn from an outside point always touch the circle at the same distance. This creates a reflective situation with two congruent right angled triangles.
a = 41°b = 49°c = 49°
41°a
bc
EXERCISES
What is the size of each angle? Copy each picture.
54°
a
bc
31°
fe
d
ih
24°g
39°j
kl
Circle Geometry
Rule 9 : ANGLE IN THE OPPOSITE SEGMENT IS EQUAL
The angle between the tangent and a chord is equal to the opposite angle in the triangle in the other segment. This is the most difficult to remember! a = 77°
b = 71°c = 32°
a
a
b
b
71°77°
ab
c
EXERCISES
What is the size of each angle? Copy each picture.
80°
68°
a
b
c
39°
89°
g
ih
66°68°
ab
c45°f
ed91°
