THE HINGE THEOREM
Standards
G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.
G.SRT.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles
Essential Question(s): What is the Hinge Theorem? How can we solve triangles when there are two possible triangles?
From your previous math experience you know that the measures of two sides and a non-included angle will not
necessarily work together to create a triangle. The Hinge Theorem is a geometric theorem that focuses on this idea.
You may have explored this idea when you studied congruent triangles.
Consider the two triangles below. Given sides of 7 cm and 4.2 cm with a non-included angle of 30, there are two
triangles that can be created. This is why angle-side-side is not a congruency theorem for triangles.
In trigonometry we consider this to be the ambiguous case for solving triangles. Looking at the two triangles above,
why do you think this theorem might be called the Hinge Theorem?
What is the relationship between the measure of ABC in the first figure and mABC in the second figure?
Consider triangle ABC to the right. Note that
mA = 30˚, AC = 7 cm and BC = 3.5 cm.
Are there still 2 possible triangles? Explain your answer.
Redraw triangle ABC so that mA = 30˚, AC = 7 cm and BC = 9 cm. Are there still 2 triangles? Explain your answer.
Redraw triangle ABC so that mA = 30˚, AC = 7 cm and BC = 2 cm. Are there still 2 triangles? Explain your answer.
1. Order the sides of the triangle to the right from longest to
shortest. (The figure is NOT drawn to scale.
2. How did you decide the order of the sides?
