5.9: Possible Triangles with SideSideAngle
Your team has just won the flag in a flag football tournament at your school. As a reward, you get to take home the flag and keep it until the next game, when the other team will try to win it back. The flag looks like this:
It makes an isosceles triangle. You start to wonder how many different possible triangles there are for different lengths of sides. For example, if you make an oblique triangle that has a given angle greater than ninety degrees, how many ways are there to do this? Can you determine how many different possible triangles there are if the triangle is an isosceles triangle?
By the end of this Concept, you'll be able to determine the answer to this question, as well as for a variety of other triangle leg lengths.
Watch This
Determining the Amount of Triangles Resulting from an SSA Triangle
Guidance
In Geometry, you learned that two sides and a nonincluded angle do not necessarily define a unique triangle. Consider the following cases given
Case 1: No triangle exists
In this case
Case 2: One triangle exists
In this case,
Case 3: Two triangles exist
In this case,
Case 4: One triangle exists
In this case
Case 5: One triangle exists
In this case,
Case 3 is referred to as the Ambiguous Case because there are two possible triangles and two possible solutions. One way to check to see how many possible solutions (if any) a triangle will have is to compare sides
If:  Then:  

a. 

No solution, one solution, two solutions 
i. 

No solution 
ii. 

One solution 
iii. 

Two solutions 
b. 

One solution 
c. 

One solution 
Example A
Determine if the sides and angle given determine no, one or two triangles. The set contains an angle, its opposite side and the side between them.
Solution:
Example B
Determine if the sides and angle given determine no, one or two triangles. The set contains an angle, its opposite side and the side between them.
Solution: Even though
Example C
Determine if the sides and angle given determine no, one or two triangles. The set contains an angle, its opposite side and the side between them.
Solution: Even though
Vocabulary
Side Side Angle Triangle: A side side angle triangle is a triangle where the length of two sides and one of the angles that is not between the two sides are known quantities.
Guided Practice
1.Determine how many solutions there would be for a triangle based on the given information and by calculating
2. Determine how many solutions there would be for a triangle based on the given information and by calculating
3. Determine how many solutions there would be for a triangle based on the given information and by calculating
Solutions:
1.
2. \begin{align*}A = 42.3^\circ, a = 16, b = 26\end{align*} \begin{align*}16 < 17.5\end{align*} 0 solutions
3. \begin{align*}A = 47.8^\circ, a = 13.48,b = 18.2\end{align*} \begin{align*}13.48 = 13.48\end{align*} 1 solution
Concept Problem Solution
As you now know, when two sides of a triangle with an included side are known, and the lengths of the two sides are equal, there is one possible solution. Since an isosceles triangle meets these criteria, there is only one possible solution.
Practice
Determine if the sides and angle given determine no, one or two triangles. The set contains an angle, its opposite side and another side of the triangle.
 \begin{align*}a = 6, b = 6, A = 45^\circ\end{align*}
 \begin{align*}a = 4, b = 7, A = 115^\circ\end{align*}
 \begin{align*}a = 5, b = 2, A = 68^\circ\end{align*}
 \begin{align*}a = 7, b = 6, A = 34^\circ\end{align*}
 \begin{align*}a = 5, b = 3, A = 89^\circ\end{align*}
 \begin{align*}a = 4, b = 4, A = 123^\circ\end{align*}
 \begin{align*}a = 6, b = 8, A = 57^\circ\end{align*}
 \begin{align*}a = 4, b = 9, A = 24^\circ\end{align*}
 \begin{align*}a = 12, b = 11, A = 42^\circ\end{align*}
 \begin{align*}a = 15, b = 17, A = 96^\circ\end{align*}
 \begin{align*}a = 9, b = 10, A = 22^\circ\end{align*}
 In \begin{align*}\triangle ABC\end{align*}, a=4, b=5, and \begin{align*}m\angle A=32^\circ\end{align*}. Find the possible value(s) of c.
 In \begin{align*}\triangle DEF\end{align*}, d=7, e=5, and \begin{align*}m\angle D=67^\circ\end{align*}. Find the possible value(s) of f.
 In \begin{align*}\triangle KQD\end{align*}, \begin{align*}m\angle K=20^\circ\end{align*}, k=24, and d=31. Find \begin{align*}m\angle D\end{align*}.
 In \begin{align*}\triangle MRS\end{align*}, \begin{align*}m\angle M=70^\circ\end{align*}, m=44, and r=25. Find \begin{align*}m\angle R\end{align*}.
Image Attributions
Description
Learning Objectives
Here you'll learn how to determine the number of solutions for triangles where two sides and the nonincluded angle are known.