Proof
Let
O
be the centre of the circle and join
CA
,
CO
, and
CB
.
Suppose
∠
ACO
=
x
.
Since
AO
,
BO
and
CO
are radii, then
AO
=
BO
=
CO
.
Since
AO
=
CO
, then
4
ACO
is isosceles, so
∠
CAO
=
∠
ACO
=
x
.
Therefore,
∠
COA
=
π

∠
ACO

∠
CAO
=
π

2
x
.
Also,
∠
COB
=
π

∠
COA
=
π

(
π

2
x
) = 2
x
.
Since
CO
=
BO
, then
4
BCO
is isosceles, so
∠
BCO
=
∠
CBO
.
Thus, looking at the angles in
4
COB
, 2
x
+ 2
∠
BCO
=
π
, so
∠
BCO
=
π
2

x
.
Therefore,
∠
ACB
=
∠
ACO
+
∠
BCO
=
x
+
π
2

x
=
π
2
, as required.
A
C
B
O
x
Compound Statements
If
A
and
B
are mathematical statements, we often see compound statements such as
“
A
and
B
”
“
A
or
B
”
For “
A
and
B
” to be TRUE, both
A
and
B
must be TRUE.
Otherwise (when one is FALSE or both are FALSE), “
A
and
B
” is FALSE.
For “
A
or
B
” to be TRUE, either or both of
A
and
B
must be TRUE.
Otherwise (when both are FALSE), “
A
or
B
” is FALSE.
Example
A
=“2 is a prime number”,
B
=“5 is a perfect square”
Is “
A
and
B
” TRUE or FALSE?
Is “
A
or
B
” TRUE or FALSE?
Aside Regarding Sets
Recall that if
A
and
B
are sets, then
A
∪
B
is the set of elements that are in either
A
or
B
, and
A
∩
B
is the set of elements that are in both
A
and
B
.
So