Building One Triangle

The triangle is one of the most important shapes in geometry, because it's present just about everywhere.  Most structures rely on triangles to keep standing, that's why building skeletons and scaffoldings often contain diagonal beams.

More important than using a triangle is finding the exact triangle you need.  A triangle consists of three sides and three angles, a total of six components. 

If you know the measures of all six components, identifying the triangle is easy.  But you don't always have all six measurements at your disposal.

You can usually find the triangle you need if you only have three measurements.  However, not every combination of three measurements will give you exactly one triangle.  Sometimes you might end up with two possible triangles, infinitely many possibilities, or no possible triangle at all.

Here are the different combinations of three components and whether they produce exactly one triangle:

Name	Description	One Triangle?	Restrictions
SSS	All three sides	Yes	The longest side must be shorter than the other two combined.
SAS	Two sides and the angle in between	Yes	None
SSA	Two sides and an angle not in between	It depends	See below.
ASA	Two angles and the side in between	Yes	The two angles must have a combined measure of less than 180°.
AAS	Two angles and a side not in between	Yes	The two angles must have a combined measure of less than 180°.
AAA	All three angles	No	The triangle could be of any size. You need at least one side length to fix it.

The Ambiguous Case

The SSA situation is often called the "ambiguous case," because it could result in two triangles, one triangle, or no triangle at all.

SSA is only guaranteed to work if the given angle is a right angle, and the side across from the angle is the longer of the two.  Then the situation becomes Hypotenuse-Leg (HL) and will produce exactly one triangle.

Practice

Will the following measurements produce exactly one triangle?

1.  a = 5, b = 9, c = 12

2.  a = 10, b = 14, C = 50^o

3.  a = 7, B = 98^o, C = 86^o

4.  a = 8, A = 47^o, B = 110^o

5.  A = 59^o, B = 72^o, C = 49^o

6.  a = 11, b = 14, A = 75^o

7.  a = 4, b = 10, c = 15

8.  a = 6, b = 11, B = 90^o

Answers

1.  yes

2.  yes

3.  no

4.  yes

5.  no

6.  no

7.  no

8.  yes