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"What's in a triangle? That which we call a three-sided polygon
By any other name would look as such."
–Euclid, Geomet-Romeo and Juli-Ometry
In this particular quote, Euclid (who wrote plays about shapes in Elizabethan English when he wasn't studying geometry) asks what makes the name "triangle" so special. After all, we can call a triangle anything we want and it would still be the same shape, right?
While that's true, these particular polygons are called triangles for very good reasons. If we split up the name into "tri-" and "-angle," the meaning of the word becomes very clear. The "tri-" is for three (like tripod or tricycle) and "-angle" is for, well, angle.
What's so special about these angles, though? Well, one of the most important properties of triangles is the fact that the sum of the three inner angles is always 180°. We call this little statement the Angle Sum Theorem for triangles.
We aren't gullible enough to believe everything we hear. Just because someone claims they saw some fairies doesn't mean we should buy out all the butterfly nets in the state to try and catch these mythical winged sprites. No, we'll need solid proof, and while we can't prove anything about fairies, elves, or leprechauns, we can fashion some sort of true statement about triangles and their angles. If seeing is believing, then the first thing we want to do is draw a picture of a triangle.
As pretty as it is, this picture doesn't tell us a whole lot. If we draw a line parallel to AC through the point B, that might tell us a bit more about what we're looking at. Let's label this parallel line segment DE.
We just learned a whole boatload of stuff about parallel lines, so this should help us. We know that alternate interior angles are congruent, so that means ∠CAB ≅ ∠DBA and that ∠ACB ≅ ∠CBE. So let's label that in our picture.
We can see that the total number of degrees in the triangle equals the number of degrees in ∠DBE. Since m∠DBE = 180°, we can say that the interior angles of any triangle will always add up to 180°.
Not convinced? We could write a formal proof to complete the Angle Sum Theorem with the triangle and a parallel line above it as our given data. But why should we do all the work?
Fill in the missing reasons and statements of the following proof that all interior angles of a triangle add up to 180°. The image below is our given data.
|1. ∠CAB ≅ ∠DBA||1. Alternate Interior Angle Theorem|
|2. ∠ACB ≅ ∠CBE||2. Alternate Interior Angle Theorem|
|3. m∠DBA + m∠ABE = 180||3. ?|
|4. m∠ABE = m∠ABC + m∠CBE||4. ?|
|5. ?||5. Substitution Property (4 into 3)|
|6. m∠CAB = m∠DBA||6. Definition of Congruence (1)|
|7. m∠ACB = m∠CBE||7. Definition of Congruence (2)|
|8. ?||8. Substitution Property (6 and 7 into 5)|
|9. Sum of interior angles is 180.||9. Interpretation of 8 into words|
This may seem like a lot at first glance, but we can take it one step at a time. The first two steps are done for us, so let's move on to the third.
What allows us to say that m∠DBA and m∠ABE equal 180°? Well, they're supplementary angles. The fact that they add up to 180° is part of their definition. All we have to write is, "Definition of Supplementary Angles."
The next statement says that m∠ABE = m∠ABC + m∠CBE. Both ∠ABC and ∠CBE are adjacent, so we can add them up according to the Angle Addition Postulate.
Substituting the fourth statement into the third means we replace "m∠ABE" with "m∠ABC + m∠CBE." That means our statement is "m∠DBA + m∠ABC + m∠CBE = 180."
We can set the measures of congruent angles equal to each other according to the definition of congruence. Actually, we already did that. All that's left is to substitute those statements into the fifth statement. If we do that, we get "m∠CAB + m∠ABC + m∠ACB = 180." We can make the final claim because ∠CAB, ∠ABC, and ∠ACB are all the interior angles of the given triangle.
This theorem will be super important to us. Like Proactiv, it'll work no matter how beautiful or how awkward our triangles are. The Angle Sum Theorem won't give us clear skin, but it will give us clearly defined angles.
Now that we know it isn't lying to us, we can use the Angle Sum Theorem to find the measures of some angles. Let's get to it!
A triangle has angles of 73° and 48°. What is the measurement of the final angle?
We know that all the angles in a triangle total to 180°, so we can set up the equation m∠1 + m∠2 + m∠3 = 180. Since we already know the measures of two angles, we can substitute them into the equation to find the measure of the third angle.
73 + 48 + m∠3 = 180
m∠3 = 59
That means the final angle in the triangle is 59°.
The angles inside of a triangle (those angles that have to add up to 180°) are called interior angles. 'Cause they're on the interior of the triangle. Duh.
Find the measure of x.
We can use the Angle Sum Theorem to find x, but for that to work, we have to find the other angles of the triangle. Luckily, we can do that.
One of the angles is already marked as 37°. One down; two to go.
We aren't given any other direct measurements, but another interior angle in the triangle has a supplementary angle of 113°. Since we know supplementary angles add up to 180°, the other angle has a measure of 180° – 113° = 67°.
All angles in a triangle add up to 180° (thanks, Angle Sum Theorem), so we can add the angles up to find x.
37 + 67 + x = 180
x = 76
We could calculate all the angles in a triangle, or we could use a special property about these things called exterior angles.
In this triangle, ∠d is the exterior angle. An exterior angle is what you get when you extend one of the sides of the triangle. The two angles not adjacent to an exterior angle, in this case ∠a and ∠b, are called remote interior angles (even though they're pretty close by).
Since an exterior angle and its adjacent interior angle are supplementary, they add up to 180°. According to the Angle Sum Theorem, angles ∠a, ∠b, and ∠c also add up to 180°. Since m∠c + m∠d = 180, and m∠a + m∠b + m∠c = 180, we can set them equal to each other.
m∠c + m∠d = m∠a + m∠b + m∠c
We have m∠c on both sides, so we can eliminate it.
m∠d = m∠a + m∠b
This is called the Exterior Angle Theorem, which says that the measure of an exterior angle is equal to the sum of the measures of the remote interior angles. It's useful just because it prevents us from having to do even more work.
And speaking of useful, here's another sweet tidbit of info: all three of a triangle's exterior angles will always, always, always add up to 360°, a.k.a double the sum of its interior angles. Wild, right?
Studying triangles is exactly what it sounds like—studying three angles. If that's the case, we'd better go back (to the future?) and dust some cobwebs off those angle concepts we learned a few chapters ago. Here's a quick refresher:
Why have these angles come back to bite us in the bum? Well, that's how we classify triangles: based on their angles. For example, an acute triangle is a triangle whose three interior angles are all acute.
An obtuse triangle is a triangle that has one obtuse angle.
Fitting more than one obtuse angle into a triangle would be like fitting more than one elephant into a New York City apartment. It just can't be done.
We say that a triangle is a right triangle if it's politically conservative. Or if one of its interior angles is a 90° angle.
We call the side that is opposite of the right angle the hypotenuse (pronounced "hai-PAW-teh-noose"). We call the other two sides the legs.
What kind of triangle is this?
Just looking at it, we can see that one of the angles is 90° and the other two are acute. If we have acute angles and a right angle, what sort of triangle is this? Acute triangles are an all-or-nothing deal. If even one angle is 90° or above, it's not an acute triangle. Since this one has a 90° angle, it's a right triangle.
Since all triangles have angles, they can all be classified as acute, obtuse, or right. We just did that. On the other hand, triangles aren't made of only angles. They also have sides that we can classify. That means we can also distinguish triangles based on their relative side lengths.
For instance, an equilateral triangle is a triangle whose three sides are equal in length. Wanna construct one? We've gotcha covered.
We don't call these triangles "equilateral" because we want the other triangles to beat them up during lunch. (Honestly, who names their kid Equilateral, anyway?) We call them equilateral because all their side lengths equal each other, but also because all their angle measures equal each other (that's called being equiangular). We'll prove this a little later, but for now, you'll have to trust us.
A triangle that has two sides of equal length is called an isosceles triangle. We can get all sorts of isosceles triangles. There are acute isosceles triangles, obtuse isosceles triangles, and right isosceles triangles. They all have different angles, but since isosceles refers to sides only, they're all are equal in terms of their isosceles-ness.
If each side of a triangle goes its own way in terms of side length, we say that it's a scalene triangle. So many triangles are scalene that mathematicians just assume triangles are scalene unless stated otherwise. That's called being scalene-normative, but since we're studying math too, we'll succumb to peer pressure.
What would be the best description of ∆ABC if ∠ACD = 79°?
We can describe triangles by their angles and by their sides, and a good place to start is the triangle itself. That means we look at the picture.
We don't know about AB, but the tick marks on AC and CB mean that those two sides are equal in length. Two equal sides means isosceles, so what we have here is an isosceles triangle.
If we take a quick look at the angles, we can be even more specific in our description of ∆ABC. Since exterior angle ∠ACD is 79°, we know that interior angle ∠ACB has a measure of 180° – 79° = 101°. An interior angle over 90° is all we need for an obtuse triangle. That means ∆ABC is an obtuse isosceles triangle.
What would be the best description of ∆KLM?
To best describe this triangle, it's best to look at its angles and its sides. Each side of this triangle has a different number of tick marks. Like snowflakes or hipsters, each one is unique and different from the rest (well, maybe not hipsters). Even though we don't have to say it 'cause we're all scalene-normative, ∆KLM is scalene.
What about its angles? We know two of the triangle's exterior angles. We can get by with a little help from our friends, supplementary angles. Since we know ∠JKL = 123°, we also know that ∠LKM = 180° – 123° = 57°. One acute angle isn't enough to say anything conclusive, so we'll continue.
If ∠LMN = 153°, then its supplement, ∠KML, equals 180° – 137° = 43°. Another acute angle. That means it all depends on the final angle.
Since we now know two of the three interior angles of the triangle, the Angle Sum Theorem for triangles allows us to find the last one.
m∠LKM + m∠KLM + m∠KML = 180
57 + m∠KLM + 43 = 180
m∠KLM = 80
So our triangle has angles of 43°, 57°, and 80°. All of 'em are under 90°, so they're all acute. Three acute angles translate to a for-sure acute triangle.