-
If SSS
- Given sides a, b, and c,
Use the Law of Cosines to determine
m
A.
- Use the Law of Cosines to determine
mB.
- Use the sum of the angles of a triangle =
180
to find
mC.
-
If SAS
- Given sides a and b,
and
C,
Use the Law of Cosines to determine side c.
- Use the Law of Cosines to determine
B.
- Use the sum of the angles of a triangle =
180 to find
mA.
-
If ASA
- Given
mA and
mB
and side c,
Use the sum of the angles of a triangle =
180 to find
mC.
- Use the Law of Sines to determine side b.
- Use the Law of Sines to determine side a.
-
If AAS
- Given
mA and
mB
and side a,
Use the sum of the angles of a triangle =
180 to find
mC.
- Use the Law of Sines to determine side b.
- Use the Law of Sines to determine side c.
-
If SSA (Ambiguous Case)
- Given sides a and b,
and
A,
Use the Law of Sines to solve for sin
B.
-
If sinB > 1,
There is no triangle.
-
If sinB
1,
Determine mB in quadrant I.
-
If mA +
mB
180
There is no triangle.
-
If mA +
mB
< 180
There is at least one triangle.
- Determine mB
in quadrant II.
It has the same sine value
as
B .
Call this angle, B'.
- Determine mA
+ mB'
- If mA
+ mB'
180
There is only one triangle.
- Determine mC
using the sum of the angles in a triangle =
180
- Determine side c using the Law of Sines.
- If mA
+ mB'
< 180
There are two triangles.
- Determine mC
using the sum of the angles in a triangle =
180
- Determine side c using the Law of Sines.
- Determine mC'
using the sum of the angles in a triangle =
180
- Determine side c' using the Law of Sines.
- Given sides a, b, and c,
Use the Law of Cosines to determine mA.
- Use the Law of Cosines to determine
mB.
- Use the sum of the angles of a triangle =
180 to find
mC.
- Given sides a and b,
andC,
Use the Law of Cosines to determine side c. - Use the Law of Cosines to determine
B.
- Use the sum of the angles of a triangle =
180 to find
mA.
- Given
mA and
mB
and side c,
Use the sum of the angles of a triangle = 180 to find
mC.
- Use the Law of Sines to determine side b.
- Use the Law of Sines to determine side a.
- Given
mA and
mB
and side a,
Use the sum of the angles of a triangle = 180 to find
mC.
- Use the Law of Sines to determine side b.
- Use the Law of Sines to determine side c.
- Given sides a and b,
andA,
Use the Law of Sines to solve for sinB.
-
If sinB > 1,
There is no triangle. -
If sinB
1,
Determine mB in quadrant I.
-
If mA +
mB
180
There is no triangle. -
If mA +
mB
< 180
There is at least one triangle.- Determine mB
in quadrant II.
It has the same sine value
asB .
Call this angle,B'.
- Determine mA
+ mB'
- If mA
+ mB'
180
There is only one triangle.
- Determine mC
using the sum of the angles in a triangle =
180
- Determine side c using the Law of Sines.
- Determine m
- If mA
+ mB'
< 180
There are two triangles.
- Determine mC
using the sum of the angles in a triangle =
180
- Determine side c using the Law of Sines.
- Determine mC'
using the sum of the angles in a triangle =
180
- Determine side c' using the Law of Sines.
- Determine m
- If m
- Determine m
-
If m
-
If sin
The analysis of the Ambiguous Case was taken from a letter to The Mathematics Teacher by Carolyn J. Case, Vincennes University, Vincennes, IN.