Part III Trigonometry

Chapter 8

Triangle Basics

Why Trigonometry?

Don’t know much trigonometry…

But I do know one and one is two,

And if this one could be with you,

What a wonderful world this could be.

  • Sam Cooke, “What a Wonderful World”.

Etymologically, trigonometry means “triangle measurement” (from ancient Greek). But why should triangles have their own special niche in the mathematics curriculum? Is this the work of the powerful triangle lobby? It is not. Triangles are fundamental. They are the building blocks from which all other polygons are assembled. To analyze an octagon, we do not need “octagonometry”. Instead, we triangulate the octagon (as in the figure at right), and then study the triangular pieces with trigonometry. Once we understand triangles, we understand all polygons.

Trigonometry is also concerned with circles. This is because triangle measurement involves angle measurement, which itself is inherently circular: $ 360^{} $ is a full circular rotation, while any smaller angle represents a portion of a circular rotation. Trigonometry’s circular aspects will appear in later chapters. For now, I’ll just note that circular rotation – a point endlessly orbiting a circle’s center – is taken as the fundamental example of periodic behavior (continuous repetition) in science and mathematics. $ ^{*} $ Trigonometry’s link with periodicity makes it indispensable in the study of rotations, vibrations, and wave motions, including sound waves and light waves.

Special Pairs (and Trios) of Angles

Certain pairs of angles have special names. Angles whose sum is \(90^{\circ}\) are complements. Angles whose sum is \(180^{\circ}\) are supplements. (Thus, the complement of \(40^{\circ}\) is \(50^{\circ}\). The supplement of \(40^{\circ}\) is \(140^{\circ}\).)

When two lines cross (as in the letter X), the resulting nonadjacent angles are called vertical angles. $ ^{*} $

Vertical angles are clearly equal to one another.

If we define a “zigzag” as three line segments joined end to end, two of which are parallel, then zigzag angles are equal. The equality of zigzag angles is obvious to most people (so obvious that instead of proving it, I’ll just appeal to your intuition here), but a few readers might be interested to know that, in fact, deep logical waters run underground precisely at this spot. $ ^{†} $

Finally, what I call flag angles (see the figure) are always equal to one another. A “flag” is the result of two parallel lines emerging from different points on the same line (the “flagpole”). As with constellations, some imagination is needed to see the flag.

Now let’s consider a trio of angles. Suppose that tomorrow, a drunk blindfolded Australian will throw three darts at a wall. This event lies in the future, yet I can tell you now (and from the other side of the Earth) the sum of the three angles in the triangle that will be formed by his darts’ tips. What black magic is this? Yes, everyone and his mother “knows” that every triangle has an angle sum of $ 180^{} $ , but few understand why this is true. You will be one of the proud few after you’ve read and digested the following retelling of Euclid’s own proof (Elements I.32). I hope you enjoy this 2300 year old work of art.

Theorem. In every triangle, the angle sum is $ 180^{} $ .

Proof. If a triangle’s angles are $ ,,$ , we must show that $ ++^{} $ . To this end, extend the triangle’s base past one of its vertices, and draw a line through that vertex parallel to the triangle’s opposite side.

The angles marked $ $ in the figure are equal because they are zigzag angles.

The angles marked $ $ are equal because they are flag angles.

We now see that angles $ ,, $ and $ $ , when put next to one another, make up a straight line. Since, by definition, a “straight angle” is $ 180^{} $ , it follows that $ ++^{} $ , as claimed.

Incidentally, you can feel this famous theorem’s truth in a very tangible way: Cut out a paper triangle, tear off its corners, and… rearrange them to form a straight line.

Exercises

    1. In the figure at right, suppose we know that the three horizontal lines are parallel. What (if anything) can we conclude about the twelve marked angles? Explain your reasoning, using the vocabulary introduced above.
  1. If we do not know if the three horizontal lines are parallel, what (if anything) can we conclude about the marked angles? Again, explain your reasoning.

  1. Do all quadrilaterals have the same angle sum? If so, prove it. If not, why not?

[Hint: Draw a diagonal – that is, a line segment joining two nonadjacent vertices.]

  1. Do all pentagons (i.e. five-sided polygons) have the same angle sum? If so, prove it. If not, why not?

  2. Do all n-sided polygons have the same angle sum? If so, prove it. If not, why not?

  3. A regular polygon is one in which all sides are equal and all angles are equal.

  1. How many degrees is each angle of a regular pentagon?

  2. How many degrees is each angle of a regular octagon?

  3. How many degrees is each angle of a regular chiliagon (a 1000-sided polygon)?

  4. Do you find it odd that although you cannot possibly visualize a chiliagon, you can, nonetheless, determine the precise size of its angles? Descartes was the first of many philosophers to ponder the chiliagon – which, by the way, is pronounced KILL-ee-a-gon, and has nothing to do with spicy stew.

  1. A parallelogram, by definition, is a quadrilateral whose opposite sides are parallel to one another. Prove that in all parallelograms, opposite angles are equal to one another. [Hint: See the hint for Exercise 2.]

  2. Can a triangle have two obtuse angles? If so, describe an example of such a triangle. If not, why not?

Which Data Determine a Triangle?

What are you? What am I? Nobody knows who anybody is. The data which life furnishes towards forming a true estimate of any being are as insufficient to that end as in geometry one side would be to determine the triangle.

  • Mark Winsome, from Herman Melville’s The Confidence Man, Ch. 36.

A triangle has six parts: three sides and three angles. However, knowing just three of these six parts often gives us enough information to determine the triangle’s shape and size. But for that to happen, the three parts we know must be the right parts in the right order. Various trios will do the trick, and in this section we’ll spell out exactly which ones will work. We’ll begin with the two most obvious ones.

SAS. A triangle’s shape and size are determined by two sides and the angle between them.

Proof. If we know two sides and their included angle (as in the figure at left), then clearly there is only one way to complete the triangle – namely, by joining its two free vertices. Since the triangle can be completed in only one way, its shape and size are completely determined by SAS, as claimed.

ASA. A triangle’s shape and size are determined by two angles and the side between them.

Proof. If we know two angles and their included side (see the figure at left), then it’s clear that we can complete the triangle in just one way: extend the “unknown” segments until they meet. Hence, the triangle’s shape and size are determined by ASA, as claimed.

Our next two “determining trios” are essentially consequences of the first two:

AAS. A triangle’s shape and size are determined by two angles and a side not between them.

Proof. The third angle is determined, too: It must be $ [180^{} - ()] $ . Hence, the given side lies between two known angles. Thus, the triangle is determined by ASA.

SSS. A triangle’s shape and size are determined by its three sides.

Proof. Suppose the three sides were mailed to us, loose in a box. Join two, as in the left figure,

letting their shared vertex act as a hinge. Changing this “hinge angle” changes the distance between the two free endpoints. Clearly, only one angle size makes this distance equal to the remaining side’s length, so this angle must be one of the triangles’ angles. This newly-

determined angle lies between two given sides, so the triangle is determined by SAS.

Let’s take stock. There are only six possible ordered trios of triangle parts (one trio with three sides: SSS; two with two sides: SAS, ASS; two with one side: ASA, AAS; and one with no side: AAA). Of these six, we’ve investigated four. Only AAA and ASS remain. Alas, neither of these will fully determine a triangle.

AAA is NOT enough to determine a triangle’s size and shape.

For example, the figure at right shows two triangles that have the same trio of angles, but the triangles have very different sizes.

AAA is enough to determine a triangle’s shape (as we’ll soon discuss), but we are interested here in trios that determine both shape and size, so AAA is out.

ASS is NOT enough to determine a triangle’s size and shape.

Observe that in the figure, the same “ASS trio” appears in two triangles with distinct shapes: $ ABC $ and $ ABD $ . In fact, every ASS trio in which the known angle is acute is, like this one, compatible with two distinct triangles. (Try to convince yourself of this by drawing a few pictures.) Thus, “the ambiguous ASS” (as I call it) is generally not enough to determine a triangle’s shape and size.

However, when the angle in ASS happens to be right, then ASS is enough to determine the triangle.

RASS. A right triangle is determined by its right angle, a leg, and its hypotenuse.

Proof. Knowing two sides of a right triangle, we can get the third with the Pythagorean Theorem. Thus, the triangle is determined by SSS. Thus, RASS fully determines a triangle, as claimed.

It is easiest to remember our list of determining trios negatively:

Determining Trios (a Summary).

• AAA and the ambiguous ASS do not determine a triangle’s shape and size.

  • The other trios (SSS, SAS, ASA, AAS – and the special case RASS) do.

In the next few chapters, we will build a “trigonometry machine” capable of finding the missing parts of any determined triangle. For example, if a triangle has a $ 54^{} $ angle surrounded by sides of length 2 and 3, we know already that the triangle is determined (by SAS), but our trigonometry machine will tell us more: When we use it correctly, it will tell us the actual numerical values of the triangle’s unknown parts. $ ^{*} $

Recognizing where we should apply the trigonometry machine is half the battle. For example, consider the figure at right. Suppose we know the sizes of its marked parts (two side lengths and three angles), and our problem is to find CD.

The thoughts of an experienced trigonometrist might run as follows:

$ ABC $ is determined (by SAS), so I can use trigonometry to find its side CB. Then $ BCD $ will be determined (by ASA), so trigonometry on that triangle will give me CD. Fine. I can solve this problem. Now I just need to carry out the details.

You’ll learn the details in the next few chapters. For now, here are some exercises.

Exercises

  1. Suppose you know the marked sides and angles in the figure. Suppose you also know that quadrilateral EGHF is a square. Your problem: Outline a strategy for finding GH that you could give to someone who already knows how to use trigonometry to find the missing parts of determined triangles.

  2. Same story as the previous problem, but this time, give a strategy to find the length of the semicircular arc in the lower left figure.

  1. Same story, but this time you must outline a strategy for finding angle $ $ in lower right figure.

Congruent Triangles

Two triangles are said to be congruent if they have the same shape and size. Apart from their locations, congruent triangles are identical: You can place one on top of the other, matching them up perfectly.

To show that two triangles are congruent, we just show that they share the same genetic material: the same determining trio. $ ^{*} $ For this reason, the various determining trios (SSS, SAS, ASA, AAS, and RASS) are often called congruence criteria.

In geometry, we often use the congruence criteria to prove that two lengths (or angles) are equal. The basic technique is to show that they are corresponding parts of congruent triangles.

Example. Prove that the opposite sides in a parallelogram are equal.

Proof. Consider parallelogram ABCD. Draw the diagonal AC, which splits the parallelogram into two triangles. We shall now prove that these two triangles are congruent.

The triangles have two angles in common (the zigzag angles $ AD = CB $ and $ DC = AB $ ) and they literally share the side between them: AC. In other words, the triangles share the same determining trio (ASA). It follows that the triangles are congruent.

This is good news, since we know that corresponding parts of congruent triangles are equal. In particular, the sides opposite the angles I’ve marked with one arc are equal (that is, AD = BC). Similarly, the sides opposite the angles I’ve marked with two arcs must be equal, so DC = AB. We’ve now proved that the parallelogram’s opposite sides are indeed equal, as claimed.

Finally, some notation: The symbol $ $ is often used to mean “is congruent to”. For example, in the preceding problem, we established that $ ABC CDA $ (by ASA).

Exercises

  1. A rhombus is, by definition, a quadrilateral whose four sides are equal. Draw some rhombi. Must the opposite angles in every rhombus be equal? If so, prove it. If not, give a counterexample.

[Hint: Reread the short paragraph before the preceding example.]

  1. A diagonal of a regular pentagon is a line segment joining any two nonadjacent vertices. Prove that each angle of a regular pentagon is trisected (cut into three equal parts) by the two diagonals that we can draw through it.

[Hints: In an isosceles triangle, the angles opposite the equal sides are equal. Also, recall exercise 5a.]

  1. Just as there are various criteria for congruence, so there are various criteria for parallelism. Here’s one you probably know: If the two angles in a Z-shape are equal, then the Z’s “top” and “bottom” are parallel. Use this parallelism criterion to prove that every rhombus is a parallelogram.

[Hint: How will you show those angles are equal? Remember the hint from exercise 11.]

  1. Is every parallelogram a rhombus? Is every square a rhombus? Is every rhombus a square? Is every square a parallelogram? How do you know?

Similar Triangles

Triangles with the same shape (i.e. the same proportions) are said to be similar triangles. For example, if you enlarge a triangle on a computer screen, the enlarged triangle will be similar (but not congruent) to the original. There are various similarity criteria for triangles, but we’ll need only one in this course: AAA.

AAA Similarity. If two triangles have the same angles, they are similar.

Proof. In the figure at right, if the dotted lines are parallel, then the two triangles (one inside the other) must be similar. Most people find this obvious after a little thought, and I’m going to assume that you will too. $ ^{*} $

Now suppose $ ABC $ and $ A{}B{}C^{} $ are two triangles with the same angles. Superimposing them so that angles A and $ A^{} $ coincide leads to a figure like the one above. Thus, to establish our triangles’ similarity, it suffices to show that BC and $ B{}C{} $ are parallel. But these lines must be parallel since they meet the same line $ (BB^{}) $ at equal angles. Hence, the triangles are similar, as claimed.

The AAA similarity criterion could be called the AA similarity criterion, since if triangles have two angles $ ( ) $ in common, then they automatically have all three angles in common; in both triangles, the third angle must be $ 180^{} - (+ ) $ . We shall use the symbol $ $ to mean “is similar to”.

The proportionality of similar figures is precisely what allows us to extract information from them. You’ve surely used similar triangles already in simple problems like this:

Problem. Find the length of side x in the figure.

Solution. The triangles are similar (by AAA), so they have the same proportions. In particular, x/3 = 10/4.

Solving this yields x = 15/2.

You may also have seen this pretty little result, which we’ll put to surprising use in the next section.

Claim. In any right triangle, if we drop a perpendicular from the right angle to the hypotenuse, we obtain two subtriangles that are similar to one another… and to the original triangle.

Proof. Let $ ABC $ be a right triangle with a right angle at C, and with angles $ $ and $ $ at vertices A and B respectively. Drop a perpendicular from C to AB, and call its foot D.

By the AA similarity criterion, we know that $ ACD ABC $ (both triangles contain a right angle and $ $ ) and $ CBD ABC $ (both triangles contain a right angle and $ $ ). Thus, we know that both subtriangles are similar to the original triangle. Moreover,

two triangles that are similar to a third must obviously be similar to one another, so all three triangles are similar, as claimed.

Exercises

  1. A quick notation reminder.

  1. What does $ ABC DEF $ mean?

  2. What does $ ABC XYZ $ mean?

  1. There are 10 triangles in the figure at right.
  1. Are they all similar to one another? Why or why not?

  1. Suppose we wish to extend the pattern even further towards vertex B. How many more similar triangles can we pack this way into $ D_{7}D_{8}B $ ?
  1. Drawing all five diagonals of a regular pentagon yields a pentagram, a figure in which ancient Pythagoreans, medieval Christians, and modern occultists of various stripes have discerned mystical significance. $ ^{*} $ You can explore some of its fascinating geometrical properties here. As you do, don’t forget the result that you established in exercise 12.

  1. Prove that $ EDC EA’C $ . How many congruent copies of this triangle appear in the pentagram?

  2. Prove that $ CDEA’ $ is a rhombus. How many congruent copies of it appear in the pentagram?

  1. Prove that $ EDC AA’B $ . How many triangles of this shape appear in the pentagram?

  2. Prove that $ EDE’ $ is isosceles. [Hint: If two angles in a triangle are equal, so are the sides opposite them.] How many congruent copies of this triangle are in the pentagram? How many similar copies?

  3. If each side of the original pentagon is 1, how long is each diagonal? [Hint: Call the diagonal d. Use the results of the previous two parts of this problem to set up a proportion involving d. Solve it for d.] This number, the diagonal-to-side ratio in any regular pentagon, is called the golden ratio. Read about it.

  4. Prove that the inner pentagon, $ A{}B{}C{}D{}E^{} $ , is a regular pentagon.

Exercises

  1. Prove the first PT shortcut formula: If \(a\) and \(b\) are the legs, the hypotenuse is \(\sqrt{a^{2}+b^{2}}\).

    1. Use the shortcut formula from the previous exercise to compute – in your head – the hypotenuse of each right triangle in the figure.
  1. Wasn’t that easier than having to define a new variable and solve an equation each time?

  1. Prove the other PT shortcut formula: if c is the hypotenuse, and l is a leg, the other leg is $ $ .

  2. Use the shortcut formula that you proved in the previous exercise to compute – in your head – the missing leg in each of the three right triangles in the figure at right.

  1. Now that you know the two PT shortcuts, use them to find – in your head – the missing sides of the following right triangles:

  1. Make up some more problems like those in the previous problem and solve them. Check your answers by solving the same problems “the long way” (i.e. by introducing a symbol for the unknown side, setting up an equation via the PT, and finally, solving the equation for the unknown.)

    1. What are the angles in a right isosceles triangle?

  1. Draw some right isosceles triangles to get a feel for what they look like.

  2. Prove that in any right isosceles triangle, the hypotenuse is $ $ times as long as the legs.

Now use Part C to answer the following questions quickly, without writing anything down…

  1. The legs of a right isosceles triangle are 3 units long. How long is the hypotenuse?

  2. The legs of a right isosceles triangle are $ 7 $ units long. How long is the hypotenuse?

  3. The hypotenuse of a right isosceles triangle is $ 4 $ units long. How long are the legs?

  4. The hypotenuse of a right isosceles triangle is 3 units long. How long are the legs?