Consider this scenario: For history class, you and a partner are building a scale model of a famous ship. It's a major semester project and will form a large percentage of your grade for the course, so you want to make sure that your historical model is as accurate as possible. In other words, the model needs to look just like the ships you've seen in historical photographs. One image, in particular, will serve as your guide when building the model.

Fortunately, many parts of the original ship, such as the sail and parts of the ship's hull, take the shape of triangles. To ensure that your model is identical to the original (though much larger than the ship in the photograph and much smaller than the real ship), you just need to verify that the corresponding triangles on the two versions are similar.

Do you remember the two criteria required to prove that triangles are similar?

If you understand the properties of similar triangles, the triangle sum theorem can help you verify that two triangles are similar even more quickly and easily. Use the slide show below to learn how.

Analyze the two triangles above. Which property of similar triangles can you use to verify that two triangles are similar? Triangles ABC and PQR are similar by the AA (angle-angle) similarity property. (Two corresponding angles are congruent.) How are the AA similarity property and the triangle sum theorem related? Can one help explain the other? Click each image below to see the steps that are required to show this relationship. Step1 Write the equations for the angles of each triangle, using the triangle sum theorem. m∠A + m∠B + m∠C = 180° m∠P + m∠Q + m∠R = 180° Step2 Next, analyze the triangles. Notice that the following angles are congruent. ∠A  \(\mathsf{ \cong }\)  ∠P ∠B  \(\mathsf{ \cong }\)  ∠Q What can you say about angles C and R? Step3 According to the triangle sum theorem, angles C and R must be congruent—if the other two corresponding angles are congruent. And if all three angles are congruent, then the sides must have a similar ratio, which makes these two triangles similar. So how could you check to see if the two triangles on your ship are similar? Think of some ideas and then click the Button below to check your thinking. It would be enough to measure just two angles on each triangle. If two corresponding angles are congruent, then the triangles are similar (built to scale). If either set of corresponding angles is not congruent, then you'll have to recreate or adjust that ship part.