April 10, 2016
Michael Sun
Angle Chasing
Angle chasing is an extremely important skill. Here is a warm-up:
Triangles
ABC
and
ADC
are isosceles with
AB
=
BC
and
AD
=
DC
. Point
D
is inside
4
ABC
,
6
ABC
= 40
◦
, and
6
ADC
= 140
◦
. What is the degree measure of
6
BAD
?
Source: AMC 12, 2007
Solution:
A
C
B
D
6
ABC
= 40 implies that
6
BAC
=
6
BCA
= 70
.
6
ADC
= 140 implies that
6
DAC
=
6
DCA
= 20
.
So
6
BAD
= 70
-
20 =
50
.
Hopefully these types of problems are pretty natural. In general, isosceles and equilateral
triangles are great for angle chasing. Next, you should be familiar with the following from
geometry:
O
B
A
C
Prove that
6
ABC
=
1
2
6
AOC
.
Proof:
As hinted above, isosceles triangles are great for angle chasing. Note that triangles
AOB
and
COB
are both isosceles.
6
ABC
=
6
ABO
+
6
OBC
=
1
2
(180
-
6
AOB
)+
1
2
(180
-
6
COB
) =
180
-
1
2
(360
-
6
AOC
) =
1
2
6
AOC.
Note that to make the proof rigorous, we also need to
1
consider the cases of the angle being acute, right, and obtuse, separately, which essentially
uses the same idea.
What’s great about this is that if you imagine sliding
B
along arc
AC
,
6
ABC
will be
constant.
A
B
C
D
E
Here, we’ll prove that
6
CEB
=
6
AED
is equal to the average of arcs
BC
and
AD
.
Proof:
6
AED
=
6
CAB
+
6
ACD
by the exterior angle theorem. And
6
CAB
+
6
ACD
=
1
2
(6.0ptAD
_
+
6.0ptBC
_
)
.
