1. Angles in Circles Objectives: B GradeUse the angle properties of a circle. A GradeProve the angle properties of a circle.

2. Angles in Circles Parts of a circle Radius diameter circumference Major Sector Major arc Minor arc Minor segment Major segment Minor Sector A line drawn at right angles to the radius at the circumference is called the Tangent

3. Angles in Circles Key words: Subtend:an angle subtended by an arc is one whose two rays pass through the end points of the arc arc two rays angle subtended by the arc end points of the arc arc end points of the arctwo rays angle subtended Supplementary:two angles are supplementary if they add up to 180 o Cyclic Quadrilateral:a quadrilateral whose 4 vertices lie on the circumference of a circle

4. Angles in Circles The angle subtended in a semicircle is a right angle Theorem 1:

5. Angles in Circles Now do these: a 70 o b 45 o c 130 o x2x2x a = 180-(90+70) a = 20 o b = 180-(90+45) b = 45 o 180-130 = 50 o c = 180-(90+50) c = 40 o 3x = 180-90 x = 30 o

6. Angles in Circles The angle subtended by an arc at the centre of a circle is twice that at the circumference Theorem 2: This can also appear like a 2a arc a 2a arc 2a2a a or

7. Angles in Circles d = 80 ÷ 2 d = 40 o e = 72 × 2 e = 144 o f = 78 ÷ 2 f = 39 o 67 ÷ 2 = 33.5 o g = 180-33.5 g = 147.5 Now do these: d 80 o f 78 o 72 o e g 67 o 96 o h h = 96 × 2 h = 192 o

8. Angles in Circles The opposite angles in a cyclic quadrilateral are supplementary (add up to 180 o ) Theorem 3: a b c d a + d = 180 o b + c = 180 o

9. Angles in Circles i = 180 – 83 = 97 o j = 180 – 115 = 65 o k = 180 – 123 = 57 o l = m = 180 ÷ 2 = 90 o 115 o i j Now do these: 83 o 123 o k l m Because the quadrilateral is a kite

10. Angles in Circles Angles subtended by the same arc (or chord) are equal Theorem 4: same arc same angle same arc same angle

11. n = 15 o p = 43 o q = 37 o r = 54 o s = 180 – (37 + 54) = 89 o Angles in Circles Now do these: 15 o p 43 o n r 54 o q 37 o s

12. Angles in Circles Summary The angle subtended in a semicircle is a right angle The angle subtended by an arc at the centre of a circle is twice that at the circumference a 2a arc The opposite angles in a cyclic quadrilateral are Supplementary (add up to 180 o ) a b c d a + d = 180 o b + c = 180 o Angles subtended by the same arc (or chord) are equal

13. More complex problems Angles in Circles 64 o a b c d e 65 o f 28 o a = 90 o The angle subtended in a semicircle is a right angle Angle AED is supplementary to angle ACD b = 180 – 64 = 116 o A B C D E O Cyclic quadrilateral ACDE Angle ABD is supplementary to angle AED b = 180 – 116 = 64 o Cyclic quadrilateral ABDE The angle subtended by an arc at the centre of a circle is twice that at the circumference d = 130 o The angle subtended by an arc at the centre of a circle is twice that at the circumference or Angles subtended by the same arc (or chord) are equal e = 65 o Opposite angles are equal, therefore triangles FGX and HXI are congruent. F G H I X f = 28 o

14. a 70 o b 45 o c 130 o x2x2x Worksheet 1 Angles in Circles a = b = c = x = d 80 o f 78 o 72 o e g 67 o 96 o h d = e = f =g = h =

15. Worksheet 2 Angles in Circles 115 o i j 83 o 123 o k l m 15 o p 43 o n r 54 o q 37 o s i = j = k=k= l = m = n = p = q = r = s =

16. Worksheet 3 Angles in Circles 64 o a b c d e 65 o f 28 o A B C D E O F H I a = b = c = d = e = f =