Transcript Document

Geometry 6
Level 1
Parts of a circle
Why is this triangle isosceles?
Two sides are radii
Find the value of x
35
x
Find the value of x
x
60
Find the value of x
x
Find the value of x
x
20
Reminder: Exterior angle equals the sum
of the opposite interior angles.
Find the value of x.
x
70
x = 70 + 50 =120
50
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
 at centre
2x
x
Find the value of x
50
x =100°
x
Find the value of x
x = 80°
160
x
Find the value of x
x = 45°
x
90
Find the value of x
x = 30°
60
x
Another way of looking at this.
2x
x
Find the value of x.
x =120°
240
x
Find the value of x.
x =125°
110
x
Find the value of x.
135
x = 90°
x
Notice if AC is a diameter
x
2x
x
Find the value of x.
x =110°
x
55
Find the value of x.
x = 40°
80
x
Find the value of x.
x = 70°
x
140
Find the value of x.
50
x = 40°
x
Find the value of x.
x = 88°
x
74
62
Find the value of x.
x = 46°
62
74
x
Find the value of x
75
x
Find the value of y
75
75
y
Find the value of z
75
75
z
75
Angles on the same arc are the same.
Angles on the same arc are the same.
Angles same arc
Why is it important?
Find A, B and C
If BC is a diameter…
x
2x
x
If BC is a diameter…
x
2y
2x
x
y
y
If BC is a diameter…
2x + 2y =180°
x
2y
2x
x
y
y
If BC is a diameter…
2x + 2y =180°
x
2y
2x
x
x + y = 90
y
y
 in a semi-circle
If AC is a diameter, find the value of
x.
50
x
x = 40°
If AC is a diameter, find the value of
x.
x = 40°
50
x
If AC is a diameter, find the value of x
x
40
x
360 - 2x
2x
180 - x
x
180 - x
Cyclic Quadrilateral
Web animation
Cyclic quadrilateral
• Opposite angles in a
cyclic quadrilateral
sum to 180 degrees.
Opposite angles add up to 180 in a cyclic
quadrilateral
Find the value of x
x = 60°
120
x
Find the value of angles A and B.
B =120°
A = 75°
Find the value of angles A and B.
110
B =110°
A
B
A = 70°
PQ is a diameter. RPQ = 30.
Find RSP.
Chord
If a diameter cuts a chord in half, it form a
right angle.
Tangent
A radius is always perpendicular to a
tangent at the point of contact.
Tgt
rad
Find the values of x, y and z
z
y
x
72
Find the values of x, y and z
x = 18°
y = 90°
z = 72°
z
y
x
72
Find the values of x, y and z
Tgt. — rad
x = 18°
y = 90°
z = 72°
z
y
 In semi-circle
’s ∆
x
72
Notice z
x = 18°
y = 90°
z = 72°
z
y
x
72
Notice z
x = 18°
y = 90°
z = 72°
z
y
z
x
72
Exterior angle at a tangent always equals the
opposite interior angle.
z = 72°
z
72
Find the values of angles x and y.
x
x = 71°
y = 62°
y
62
71
Find the angle ACB.
ACB = 48°
Find the value of x.
x
x
128
42
Find the value of x.
x = 52
x =69
128
42
Find the value of x
x =15°
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.
.
x =15°
75
75
a) Work out the following angles:
i) ACD
ii) BCA
iii) CDA
b) 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