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## High school geometry

### Course: High school geometry>Unit 5

Lesson 4: Ratios in right triangles

# Side 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.
Two right triangles both a vertical reflection of one another. Both long legs are congruent. Both angles between the long legs and the hypotenuses are congruent. A congruent symbol is between both triangles.
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.
How are the four triangles related?
Four right triangles that share the same point A and the same angle A. The triangles all have hypotenuses on the same line segment, A H. They also all have bases on the same line segment, A I. The smallest triangle, triangle A B C, has a base of eight units, a height of six units, and a hypotenuse of ten units. The second smallest triangle, triangle A D E, has a height of nine units. The second largest triangle, triangle A F G, has a height of twelve units. The largest triangle, triangle A H I, has a height of fifteen units.
The triangles are
according to the
criterion.

Measurement table
Here are those triangles again.
Four right triangles that share the same point A and the same angle A. The triangles all have hypotenuses on the same line segment, A H. They also all have bases on the same line segment, A I. The smallest triangle, triangle A B C, has a base of eight units, a height of six units, and a hypotenuse of ten units. The second smallest triangle, triangle A D E, has a height of nine units. The second largest triangle, triangle A F G, has a height of twelve units. The largest triangle, triangle A H I, has a height of fifteen units.
Complete the table of measurements relative to angle, A.
triangle, A, B, Ctriangle, A, D, Etriangle, A, F, Gtriangle, A, H, I
Opposite leg length691215
16
Hypotenuse length1015
25
Angle A37, degree37, degree37, degree37, degree
Right angle90, degree90, degree90, degree90, degree
Last angle
degree
degree
degree
degree

Now we're ready to start checking that data for patterns.
Ratio table
Complete the ratio table.
Round to the nearest hundredth.
triangle, A, B, Ctriangle, A, D, Etriangle, A, F, Gtriangle, A, H, I
start fraction, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, end fraction

What did you notice?

## Proving that the pattern works for another angle measure

Proof
Complete the proof that start fraction, A, C, divided by, B, C, end fraction, equals, start fraction, F, D, divided by, E, D, end fraction.
Right triangle A B C with Angle A being ninety degrees and angle B being the reference angle.
Right triangle D E F with Angle F being ninety degrees and angle E being the reference angle.
StatementReason
1angle, A, \cong, angle, FAll right angles are congruent.
2angle, B, \cong, angle, EGiven
3triangle, A, B, C, \sim, triangle
similarity
4start fraction, A, C, divided by, F, D, end fraction, equals, start fraction, B, C, divided by, E, D, end fractionLengths of corresponding sides of similar triangles form equal ratios.
5start fraction, A, C, divided by, B, C, end fraction, equals, start fraction, F, D, divided by, E, D, end fractionMultiply both sides by
.

Conclusion of proof
What did we prove?
What triangles did we prove it for?

## 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.
Extension 1.1
• Current
Given the measure of an acute angle in a right triangle, we can tell the ratios of the lengths of the triangle's sides relative to that acute angle.
Here are the approximate ratios for angle measures 25, degree, 35, degree, and 45, degree.
Angle25, degree35, degree45, degree
start fraction, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction0, point, 910, point, 820, point, 71
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction0, point, 420, point, 570, point, 71
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, end fraction0, point, 470, point, 71
Use the table to approximate m, angle, J in the triangle below.
Right triangle J K L with Angle K being ninety degrees and angle J being the reference angle. Side J K is seven point four units. Side J L is nine units.