CIRCLE THEOREMS Questions normally ask for the size of an angle and reasons why. In each case use the theorems you know and what is special. Any triangles made from radii will always be isosceles.

Tangent and Radius are perpendicular CIRCLE THEOREMS Tangent and Radius are perpendicular

Angle in a Semi-Circle is always 90° CIRCLE THEOREMS Angle in a Semi-Circle is always 90°

The angle at the Centre is twice that at the circumference CIRCLE THEOREMS The angle at the Centre is twice that at the circumference

Angles from the same chord are equal CIRCLE THEOREMS Angles from the same chord are equal Angles A and C are equal from the chord BD In the same way angle B and D are equal, coming from the chord AC

The opposite angles in a cyclic quadrilateral add to 180° CIRCLE THEOREMS The opposite angles in a cyclic quadrilateral add to 180° A and C add to 180 and so do B and D

Alternate Segment theorem CIRCLE THEOREMS Alternate Segment theorem Angle between the tangent and the chord (angle a on the diagram) equals the angle in the alternate(other) segment – angle b

Examples Try the following examples. The question is on one page, go to next page for a solution

CIRCLE THEOREMS Examples

CIRCLE THEOREMS Examples In Δ OAB angles A and B are equal OA=OB as radii which makes isosceles Angle OBA will then be (180-2x) ÷ 2 which is (90-x) OB and OT are perpendicular as they are radius and tangent and so Angle ABT will be 90-(90-x) which is x

CIRCLE THEOREMS Examples

CIRCLE THEOREMS Examples Draw the line AO and label angles at B and C as b and c In Δ OAC, OA=OC as radii so Δ is isosceles Angle OAC = OCA = c Angle AOC is 180 – 2c In same way in ΔOBC angle BOC is 180 – 2b Angles around a point are 360 so angle BOC=360- AOC – BOC = 360 – (180 -2b) –(180-2c) = 2b + 2c = 2 x ( b + c) = 2 x angle CAB b c

CIRCLE THEOREMS Examples (a) Work out the size of angle AOD° The diagram shows a circle centre O. A, B and C are points on the circumference.   DCO is a straight line. DA is a tangent to the circle. Angle ADO = 36° (a) Work out the size of angle AOD° (b) Work out the size of angle ABC. Give reasons for your answers

CIRCLE THEOREMS Angle OAD is 90 as OA is a radius and AD a tangent Angle AOD is 54° as OAD is a triangle and OAD is 90. Angle ABC is the angle at the circumference when AOC is at the centre and so is 27° Angle at centre theorem

CIRCLE THEOREMS Examples

CIRCLE THEOREMS Angle PQT = angle PTA = 58° Alternate segment theorem In Δ PQT, PQ=QT and so it isosceles and the angles at P and T are equal and as PQT is 58, they are (180-58) ÷ 2 which is 61 Angle POT is 90-58 = 32 because angle OAT is 90 (radius/tangent) Finally OTQ = 61-32 = 29°

CIRCLE THEOREMS Examples

CIRCLE THEOREMS AD is a chord and angles ABD and ACD are from that chord and so are equal.  ACD = 54°

CIRCLE THEOREMS Examples

CIRCLE THEOREMS ABCD is a cyclic quadrilateral and so angle BAD = 50 Angle BOD is the angle at the centre and so is 100° In the quadrilateral OBDC we now have 100 and 130 which leaves 360-230 = 130 for the other two. As OB=OD (radii), OBDC is symmetric and angle OBC and ODC are equal and must be 130 ÷ 2 = 65°

CIRCLE THEOREMS Examples

CIRCLE THEOREMS ABCD is a cyclic quadrilateral and so angle D is 52° Angle AOC is the angle at the centre and so will be twice that at the circumference and so is 2 x 52 = 104°

CIRCLE THEOREMS Examples

CIRCLE THEOREMS As no angles are known set the one you want to x BC=BD so Δ BCD is isosceles and so BCD is also x Angle C is 90 as it is the angle in a semi-circle and so angle ACD is 90 - x Because AC and BD are parallel, angles ACD and CDB are equal (alternate angles) and so x = 90 – x Solving gives x = 45°