Geometry 6
Level 1
Why is this triangle isosceles?
Two sides are radii
Find the value of x
35 x
Find the value of x
60
x
Find the value of x
x
Find the value of x
x20
Reminder: Exterior angle equals the sum of the opposite interior angles.
Find the value of x.
70 50
x
Let’s investigate
We now have two isosceles triangles
In each case their base angles are equal.
The exterior angle is twice the opposite internal angle
The exterior angle is twice the opposite internal angle
The angle at the centre is twice the angle at the circumference
Find the value of x
x
50
Find the value of x
x
160
Find the value of x
x
90
Find the value of x
x
60
Another way of looking at this.
2x
x
Find the value of x.
240
x
Find the value of x.
110
x
Find the value of x.
135
x
Notice if AC is a diameter
2x
x
x
Find the value of x.
55x
Find the value of x.
80
x
Find the value of x.
140
x
Find the value of x.
50
x
Find the value of x.
74
x 62
Find the value of x.
74x
62
Find the value of x
x75
Find the value of y
y
75 75
Find the value of z
z
75 7575
Angles on the same arc are the same.
Angles on the same arc are the same.
Why is it important?
If BC is a diameter…
x
x
2x
If BC is a diameter…
x
x
2x
y y
2y
If BC is a diameter…
x
x
2x
y y
2y
If BC is a diameter…
x
x
2x
y y
2y
If AC is a diameter, find the value of x.
50 x
If AC is a diameter, find the value of x.
50
x
If AC is a diameter, find the value of x
40
x
2x
x
360 - 2x
180 - x
Cyclic Quadrilateral
• Opposite angles in a cyclic quadrilateral sum to 180 degrees.Web animation
Cyclic quadrilateral
Opposite angles add up to 180 in a cyclic quadrilateral
Find the value of x
x120
Find the value of angles A and B.
Find the value of angles A and B.
110
A B
PQ is a diameter. RPQ = 30.Find RSP.
If a diameter cuts a chord in half, it form a right angle.
Tgt rad
A radius is always perpendicular to a tangent at the point of contact.
Find the values of x, y and z
72x
yz
Find the values of x, y and z
72x
yz
Find the values of x, y and z
72x
yz
Tgt. — rad
In semi-circle
’s ∆
Notice z
72x
yz
z
Exterior angle at a tangent always equals the
opposite interior angle.
72
z
Find the values of angles x and y.
62
x
y
71
Find the angle ACB.
Find the value of x.
x
128
x
42
Find the value of x.
x = 52
128
x =69
42
Find the value of x
Find, in terms of x, ADB, AOD. ABD
ADB = 2 x, AOD = 180 - 6 x,
ABD = 90 - 3 x
If the angle PCQ = 72, what is the reflex angle POQ ?
252
Find the angle ABC
112
a) Calculate the angle between the tangent and the line AC.
b) Calculate the angle ABC.
Determine the angle QAC in terms of x. Calculate the value of x.
.
Determine the angle QAC. Calculate the value of x.
.
75
75
a) Work out the following angles:i) ACD ii) BCA iii) CDAb) Show that the triangle EBA is isosceles.
ABC and ADE are straight lines. The diameter of the larger circle is CE.
Express in terms of x, the following angles:a) ABD b) BDC c) BAD