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Geometry 6 Level 1

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Parts of a circle

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Why is this triangle isosceles?

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Two sides are radii

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Find the value of x 35 x

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Find the value of x 60 x

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Find the value of x x

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x 20

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Reminder: Exterior angle equals the sum of the opposite interior angles.

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Find the value of x. 70 50 x

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Let’s investigate

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We now have two isosceles triangles

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In each case their base angles are equal.

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The exterior angle is twice the opposite internal angle

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The angle at the centre is twice the angle at the circumference

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at centre 2x2x x

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Find the value of x x 50

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Find the value of x x 160

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Find the value of x x 90

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Find the value of x x 60

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Another way of looking at this. 2x2x x

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Find the value of x. 240 x

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Find the value of x. 110 x

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Find the value of x. 135 x

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Notice if AC is a diameter 2x2x x x

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Find the value of x. 55 x

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Find the value of x. 80 x

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Find the value of x. 140 x

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Find the value of x. 50 x

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Find the value of x. 74 x 62

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Find the value of x. 74 x 62

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Find the value of x x 75

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Find the value of y y 75 75

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Find the value of z z 75 75

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Angles on the same arc are the same.

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Angles same arc

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Why is it important?

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Find A, B and C

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If BC is a diameter… x x 2x2x

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x x 2x2x y y 2y2y

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x x 2x2x y y 2y2y

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x x 2x2x y y 2y2y

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in a semi-circle

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If AC is a diameter, find the value of x. 50 x

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If AC is a diameter, find the value of x. 50 x

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If AC is a diameter, find the value of x 40 x

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2x2x x 360 - 2x 180 - x

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x

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Cyclic Quadrilateral Opposite angles in a cyclic quadrilateral sum to 180 degrees. Web animation Cyclic quadrilateral

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Opposite angles add up to 180 in a cyclic quadrilateral

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Find the value of x x 120

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Find the value of angles A and B.

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110 A B

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PQ is a diameter. RPQ = 30. Find RSP.

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Chord

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If a diameter cuts a chord in half, it form a right angle.

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Tangent

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Tgt rad A radius is always perpendicular to a tangent at the point of contact.

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Find the values of x, y and z 72 x y z

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Find the values of x, y and z 72 x y z

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Find the values of x, y and z 72 x y z Tgt. — rad In semi-circle ’s ∆

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Notice z 72 x y z

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Notice z 72 x y z z

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Exterior angle at a tangent always equals the opposite interior angle. 72 z

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Find the values of angles x and y. 62 x y 71

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Find the angle ACB.

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Find the value of x. x 128 x 42

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Find the value of x. x = 52 128 x =69 42

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Find the value of x

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Find, in terms of x, ADB, AOD. ABD

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ADB = 2 x, AOD = 180 - 6 x, ABD = 90 - 3 x

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If the angle PCQ = 72, what is the reflex angle POQ ? 252

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Find the angle ABC 112

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a) Calculate the angle between the tangent and the line AC. b) Calculate the angle ABC.

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Determine the angle QAC in terms of x. Calculate the value of x..

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Determine the angle QAC. Calculate the value of x.. 75

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a) Work out the following angles: i) ACD ii) BCA iii) CDA b) Show that the triangle EBA is isosceles.

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ABC and ADE are straight lines. The diameter of the larger circle is CE.

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Express in terms of x, the following angles: a) ABD b) BDC c) BAD

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