Lesson 6 More Than Right Develop Understanding

Learning Focus

Find missing sides and angles in non-right triangles.

How can I use trigonometry and the Pythagorean theorem to find the lengths of sides of non-right triangles?

Open Up the Math: Launch, Explore, Discuss

We can use right triangle trigonometry and the Pythagorean theorem to solve for missing sides and angles in a right triangle. What about other triangles? How might we find unknown sides and angles in acute or obtuse triangles if we only know a few pieces of information about them?

In the previous lesson we found it might be helpful to create right triangles by drawing an altitude in a non-right triangle. We can then apply trigonometry or the Pythagorean theorem to the smaller right triangles, which may help us learn something about the sides and angles in the larger triangle.

See if you can devise a strategy for finding the missing sides and angles for each of these triangles.

1.

Given: acute triangle $A B C$

2.

Given: acute triangle $A B C$

3.

See if you can generalize the work you have done on problems 1 and 2 by finding relationships between sides and angles in the following diagram. Unlike the previous two problems, this triangle contains an obtuse angle at $C$ . Find as many relationships as you can between sides $a$ , $b$ and $c$ and the related angles $A$ , $B$ , and $C$ .

Ready for More?

Try to find the missing sides of this obtuse triangle from the perspective of each of the three altitudes. Are the results consistent, regardless of the altitude used to find the sides of the triangle? Is it possible to use any of the three altitudes to solve the triangle and, if so, under what conditions?

Takeaways

Key strategies for solving for the missing sides of an oblique triangle:

If I know for an oblique triangle, then I can find the missing sides and angles using the Law of Sines:

Adding Notation, Vocabulary, and Conventions

Right triangles contain a right angle and two .

Acute triangles contain .

Obtuse triangles contain .

Acute and obtuse triangles are call oblique triangles, since .

Vocabulary

Lesson Summary

In this lesson, we learned how to find missing sides of oblique triangles by decomposing them into non-congruent right triangles and applying trigonometry and the Pythagorean theorem to the right triangles.

Retrieval

1.

Find the area of $△ A B C$ .

2.

Fill in the missing lengths of sides and measurements of angles in $△ D E F$ and $△ P Q R$ . Find exact lengths of sides and measurements of angles.