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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. a = 50° 40° a EXERCISES What is the size of each angle? Copy each diagram. a 45° c 65° 63° b 81° d Circle Geometry 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° 60° a EXERCISES What is the size of each angle? Copy each diagram. 59° a c 47° d 71° b 33° Circle Geometry Rule 3 : TWO RADII CREATE AN ISOSCELES TRIANGLE These triangles are constructed using two radii. Therefore they are isosceles triangles. a = 30° a EXERCISES What is the size of each angle? Copy each diagram. 67° a c 220° d 110° b 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 EXERCISES What is the size of each angle? Copy each diagram. b 67° a d c 41° 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 a b b a 50° a + b = 180° b a = 50° b = 130° EXERCISES What is the size of each angle? Copy each diagram. a 31° 43° e b f g d c 76° h 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 a a 2a 63° a a = 126° EXERCISES What is the size of each angle? Copy each diagram. c a 72° d b 79° 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 b c d a b a + c = 180° b + d = 180° 87° 79° a = 93° b = 101° EXERCISES What is the size of each angle? Copy each diagram. e a 101° 111° b 81° 89° f h 61° 72° 59° c d 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 b c EXERCISES What is the size of each angle? Copy each picture. a h i b c d g 31° 39° j e l f 24° 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 a = 77° b = 71° c = 32° b a b 71° 77° a c b EXERCISES What is the size of each angle? Copy each picture. b a 68° c h 80° 89° 66° 68° f 45° 91° d e a c b