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Geometry 6
Level 1
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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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Find the value of x
x20
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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 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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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.
2x
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
2x
x
x
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Find the value of x.
55x
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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.
74x
62
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Find the value of x
x75
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Find the value of y
y
75 75
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Find the value of z
z
75 7575
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Angles on the same arc are the same.
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Angles on the same arc are the same.
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Why is it important?
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If BC is a diameter…
x
x
2x
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If BC is a diameter…
x
x
2x
y y
2y
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If BC is a diameter…
x
x
2x
y y
2y
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If BC is a diameter…
x
x
2x
y y
2y
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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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2x
x
360 - 2x
180 - 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
x120
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Find the value of angles A and B.
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Find the value of angles A and B.
110
A B
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PQ is a diameter. RPQ = 30.Find RSP.
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If a diameter cuts a chord in half, it form a right angle.
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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
72x
yz
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Find the values of x, y and z
72x
yz
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Find the values of x, y and z
72x
yz
Tgt. — rad
In semi-circle
’s ∆
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Notice z
72x
yz
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
75
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a) Work out the following angles:i) ACD ii) BCA iii) CDAb) 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