Suppose the flight path you observed on the previous page involved slightly different cities so that it formed the triangle shown below.
Add the angles inside this triangle together—one obtuse angle and two acute angles. What is the sum of these angles?
The sum of this triangle's interior angles is 180 degrees.
If you add the interior angles of several more triangles, you will notice a solid pattern in the sums. In fact, the sum of the interior angles for any triangle will be 180 degrees. The rule that describes this pattern is the called the triangle sum theorem. Study the examples on these tabs to learn more about the evidence for this theorem—and how it can be used to solve problems.
The Equation
Equation Examples
Solving Problems
We can use the path of the plane to look at the triangle sum theorem in more detail.
According to the triangle sum theorem, the sum of the three interior angles of any triangle is equal to 180 degrees. Since this statement is a theorem—and not just a conjecture—it has been proved to be true for any triangle.
Can you write an equation for the interior angles created by the plane's flight path?
If m stands for the measure of an angle, then:
m∠A + m∠B + m∠C = 180°
Here are a few more examples of the triangle sum theorem at work. Click each triangle below to see how its angles add up to 180°.
Suppose you're the copilot for a flight between city A and city B. An hour after takeoff, the pilot must change the airplane's flight path to find a way around a brewing thunderstorm and the turbulence it has created. She asks you to find the angle that will be created by this change in course—so that she'll be ready to put the plane back on course for city B once she moves far enough away from the storm.
Can you use the triangle sum theorem to figure out the degree of the turn that the plane must make to resume course? Analyze the triangle below, which was created by the detour between point A and point B.
Use the triangle sum theorem to find the missing angle in this scenario. Then click the button below to check your answer.
According to the triangle sum theorem, m∠A + m∠B + m∠C = 180°.
If you plug in the information that you know . . .
27° + 82° + m∠C = 180°
you can solve for the missing angle.
109° + m∠C = 180°
m∠C = 180° – 109°
m∠C = 71°
The pilot will need to make a 71° turn to ensure that the airplane reaches city C.
