Triangles:

Congratulations! We officially have enough information to accurately talk about triangles in the same way that Euclid described and wrote about triangles in his book "Elements". Triangles have 2 different methods of classification. First, triangles can be classified based on the relative length of their sides. Second, triangles can be classified based on the size of their angles. Check out a great video describing the different ways of classifying triangles using this link: Triangle Classification $^{16}$

Triangle classification by side:

It is common knowledge that triangles are made up of three sides. One way that we can distinguish between types of triangles is to compare their sides to one another. Below is a graphic$^7$ that demonstrates the three different classifications of triangles by side.

As you can see, the three types of triangles distinguished by their sides are Equilateral, Isosceles, and Scalene.
Equilateral triangles are triangles that have 3 congruent sides. This means all the sides of the triangle have the same length.
Isosceles triangles have 2 congruent sides. This means that 2 of the sides of these types of triangles have the same length.
Scalene triangles have no congruent sides. This means that each side of a scalene triangle has a different length from the other sides.

Triangle classification by angle:

We can also classify triangles using the degree measures of their interior angles. Use the image$^8$ below as a reference for the 3 different classifications of triangles by degree angle measure.

As depicted in this display. The three types of triangle classification by angle are: obtuse, right, and acute.
Let's first talk about acute triangles. Acute triangles contain 3 acute angles. In other words the only angles that are used to make acute triangles are acute angles.
Next, let's talk about right triangles. These triangles have many interesting properties, including the Pythagorean theorem and applications to trigonometry. Right triangles are defined by having 2 acute angles and 1 right angle, hence the name.
Finally, obtuse triangles have 2 acute angles, and 1 obtuse angle.
Something important to remember is that all triangles have at least 2 acute angles, so just because you see a triangle with an acute angle, don't assume that it has to be an acute triangle!

Triangle Based Theorems:

Now that we know how to distinguish between different triangles, we can begin to talk about their properties. Triangles are the building block of geometry, but before we can use the valuable properties that triangles hold, we need to formally define these properties and the conditions that are necessary for their use.

Theorems of Congruence:

Euclid's proofs in "Elements" are riddled with triangles. They are used to prove the definitions of hyperbolas, parabolas, ellipses and so much more. One of the aspects of triangles that make them so useful is the ease of determining if a triangle is congruent to another triangle or not. Why do we care is one triangle is congruent to another you ask? We will discuss the answer to this question later on in the applications section of the page! For now, let's examine the theorems of congruence associated with triangles.

This theorem states that given two triangles, call one $\triangle{ABC}$ and one $\triangle{DEF}$, then if $\triangle{ABC}\cong\triangle{DEF}$ that means $\overline{AB}\cong\overline{DE}$, $\overline{BC}\cong\overline{EF}$, $\overline{AC}\cong\overline{DF}$, $\angle{A}\cong\angle{D}$, $\angle{B}\cong\angle{E}$, $\angle{C}\cong\angle{F}$.
This theorem is very useful, but how can I prove that two triangles are congruent without checking every element in the list above? The solutions to that question are the Triangle Congruence Theorems. We will cover the 4 main triangle congruence theorems, and a special fifth theorem that applies only to right triangles.
The first of the triangle congruence theorems is called SSS congruence. SSS stands for Side-Side-Side congruence. In other words, this theorem claims if all three sides on two triangles are congruent, that means the triangles themselves must be congruent. This is particularly useful because we can learn that the angles of one triangle are congruent with respect to the angles of the other triangle without knowing the angle measure of any of the angles.
The second triangle congruence theorem is called SAS. This stands for Side-Angle-Side congruence. This theorem states if two sides and the angle between the two sides of two triangles are congruent then the triangles themselves must be congruent. Therefore, we would then be able to claim that the other side and 2 angles must be congruent to one another, even if we do not know their length or angle measures.
The third triangle congruence theorem is called ASA. This stands for Angle-Side-Angle congruence. It states that if two angles and the side between those two angles on two different triangles are congruent then the two triangles are also congruent. Therefore, through our knowledge of the congruence of two angles and the side between the angles, we are able to claim subsequent congruence for all other parts of the triangles.
The fourth triangle congruence theorem is called AAS congruence. AAS stands for Angle-Angle-Side congruence. It states that if two angles are congruent, and one side that is not between the two angles is congruent then the two triangles are also congruent. This means that we can determine the congruence of all the parts of the triangles by identifying only the congruence of two angles and a side that is not between those angles.
The final of the triangle congruence theorems only applies to right triangles. This congruence theorem is called Hypotenuse-Leg. It states that if two right triangles have congruent hypotenuses and one congruent leg, then we can conclude the entire triangles are congruent.
Below is an image$^9$ that describes each of the 5 triangle congruence theorems and helps visualize their appearance.

Sum of the Angles of a Triangle Theorem:

You may have heard in your classes that the degree measures of the angles of a triangle add up to 180$^{\circ}$. This is an incredibly useful fact, and we are going to prove its truthfulness together! Below is the formal statement for this theorem.

Proof: Use the applet below to serve as a reference for our proof.

In the applet above we see $\triangle{ABC}$. This is a general triangle, meaning we could change the length of the sides, or the degree measures of the angles, or location of the points, and it wouldn't affect the validity of our proof. Now consider $\overleftrightarrow{AC}$. We know that there exists some line, call it $\overleftrightarrow{DE}$, such that $\overleftrightarrow{AC}\parallel\overleftrightarrow{DE}$ and $\overleftrightarrow{DE}$ intersects B. Now we have our two parallel lines, and we can also see that $\overleftrightarrow{AB}$ and $\overleftrightarrow{BC}$ each act as a transversal for $\overleftrightarrow{AC}$ and $\overleftrightarrow{DE}$. Since $\overleftrightarrow{AC}\parallel\overleftrightarrow{DE}$ we can use Theorems 1-4 that we discussed in the Interior and Exterior Angle Theorems section. To start, let's consider $\angle{DBA}$, $\angle{ABC}$, and $\angle{CBE}$. These three angles are supplementary, meaning their sum is 180$^{\circ}$. We know that because they are all on $\overleftrightarrow{DE}$ and cover the whole line. Using Theorem 1, we know that $\angle{DBA}\cong\angle{BAC}$, we also know that $\angle{CBE}\cong\angle{ACB}$. Since these angles are congruent we know that $m\angle{DBA}=m\angle{BAC}$ and $m\angle{CBE}=m\angle{ACB}$. Finally, we already showed that $m\angle{DBA} + m\angle{ABC} + m\angle{CBE}=180^{\circ}$, and we know $m\angle{DBA}=m\angle{BAC}$ and $m\angle{CBE}=m\angle{ACB}$ by Theorems 1 and 2, so we can substitute $m\angle{BAC}$ for $m\angle{DBA}$, and $m\angle{ACB}$ for $m\angle{CBE}$. Therefore we get that $m\angle{BAC}+m\angle{ABC}+m\angle{ACB}=180^{\circ}$. This is exactly what we were trying to prove since we know that $\angle{BAC}$, $\angle{ABC}$, and $\angle{ACB}$ are the interior angles of the triangle and we have shown that their degree angle measures sum to 180$^{\circ}$.

The Pythagorean Theorem:

By this point in your education, it is extremely likely that you have heard of, and worked with, the Pythagorean Theorem. Let's state it for our reference.

This theorem is very useful. This theorem is the basis behind how we measure distance, and is the backbone for trigonometry. It is also helpful because it can be used in both directions. What that means is that if I have a right triangle I know $a^2+b^2=c^2$, and vice versa.
It is not an overstatement to say that this may be the most well-known theorem in the United States, but have you ever stopped to wonder how it was discovered? This Theorem has hundreds of unique proofs. (In fact, even former U.S. President James Garfield created his own unique proof for this theorem!) For now, let's use the applet$^{15}$ below to discover the relationship described by the Pythagorean Theorem.

As shown in the applet, there are two squares whose side lengths are a and b.This means the area of each of the respsective squares is $a^2$ and $b^2$. When we divide the two squares we can turn them into puzzle pieces and show that the area of the two squares with areas $a^2$ and $b^2$ is equal to the area of the square with side length c. Therefore, we have found that, on a right triangle with side lengths a and b, and hypotenuse length of c, that $a^2+b^2=c^2$.

References:

7: Hernandez, D. (n.d.). Equilateral Isosceles And Scalene Triangles - Triangle, HD Png Download - kindpng. KindPNG.com. Retrieved November 1, 2022, from https://www.kindpng.com/imgv/iRbRbio\_equilateral-isosceles-and-scalene-triangles-triangle-hd-png/

8: University of Texas San Antonio (Director). (2018, April 1). Project 2. http://www.cs.utsa.edu/%7Ecs1063/projects/Spring2008/Project2/project2.html

9: Congruence Theorem (Triangles) | Chitown Tutoring. (2020, February 17). Chicago Test Prep & Tutoring. https://chitowntutoring.com/congruence-theorem-triangles/

14: Garfield’s proof of the Pythagorean theorem. (n.d.). Khan Academy. Retrieved November 1, 2022, from https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-pythagorean-proofs/v/garfield-s-proof-of-the-pythagorean-theorem

15: Boyaj, G. (n.d.). Pythagorean theorem: Interactive puzzle. Retrieved November 25, 2022, from https://en.etudes.ru/etudes/pythagorean-theorem/

16: Math Antics. (2013, October 14). Math Antics - Triangles. YouTube. https://www.youtube.com/watch?v=mLeNaZcy-hE

17: Skoledu. (2017, July 13). Pythagorean theorem. GeoGebra. https://www.geogebra.org/m/eFmmVrd8