Angles in Circles Objectives B Grade A Grade

Angles in Circles Objectives: B Grade A Grade Use the angle properties of a circle. Prove the angle properties of a circle.

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

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 two rays arc angle subtended by the arc 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

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

Angles in Circles Now do these: b = 180 -(90+45) b = 45 o a = 180 -(90+70) a = 20 o 70 o a b 45 o 180 -130 = 50 o c 130 o c = 180 -(90+50) c = 40 o 3 x = 180 -90 x = 30 o x 2 x

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

Now do these: d Angles in Circles d = 80 ÷ 2 d = 40 o 72 o h = 96 × 2 h = 192 o e = 72 × 2 e = 144 o 96 o h e 80 o g = 180 -33. 5 g = 147. 5 f = 78 ÷ 2 f = 39 o 78 o f 67 o g 67 ÷ 2 = 33. 5 o

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

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

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

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

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 2 a The opposite angles in a cyclic quadrilateral are Supplementary (add up to 180 o) a + d = 180 o b + c = 180 o a b c d arc Angles subtended by the same arc (or chord) are equal

Angles in Circles More complex problems C G 64 o e B c A 28 o a O b X D E The angle subtended in a semicircle is a right angle a = 90 o Cyclic quadrilateral ACDE Angle AED is supplementary to angle ACD b = 180 – 64 = 116 o Cyclic quadrilateral ABDE Angle ABD is supplementary to angle AED b = 180 – 116 = 64 o H 65 o f d F I 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 = 28 o

Worksheet 1 Angles in Circles a 70 o d 130 o c x c= b= a= 2 x x= 72 o e 80 o d= b 45 o 96 o h 78 o e= h= f= 67 o f g= g

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

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