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Course: Trigonometry > Unit 1
Lesson 1: Ratios in right trianglesSide ratios in right triangles as a function of the angles
By similarity, side ratios in right triangles are properties of the angles in the triangle.
When we studied congruence, we claimed that knowing two angle measures and the side length between them (Angle-Side-Angle congruence) was enough for being sure that all of the corresponding pairs of sides and angles were congruent.
How can that be? Even with the Pythagorean theorem, we need two side lengths to find the third. In this article, we'll take the first steps towards understanding how the angle measures and side lengths give us information about each other in the special case of right triangles.
This is a great opportunity to work with a friend or two. The goal of this article is to find and discuss patterns, not to spend a bunch of time calculating. Try splitting up the work so there's more time to talk about what you see!
Let's look for patterns
First, we'll collect some data about a set of triangles.
Now we're ready to start checking that data for patterns.
What did you notice?
Proving that the pattern works for another angle measure
What did we conclude?
If two right triangles have an acute angle measure in common, they are similar by angle-angle similarity. The ratios of corresponding side lengths within the triangles will be equal. So the ratio of the side lengths of a right triangle just depends on one acute angle measure.
Why will this be useful?
Before, we could use the Pythagorean theorem to find any missing side length of a right triangle when we knew the other two lengths. Now, we have a way to relate angle measures to the right triangle side lengths. That allows us to find both missing side lengths when we only know one length and an acute angle measure. We can even find the acute angle measures in a right triangle based on any two side lengths.
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