Working with Angles:

Before we dive into working with triangles, there are a few features of angles that we will have to understand. This section covers basic angle theorems and will provide the necessary information for working with the degree measure of angles in triangles.
Acute, Right, and Obtuse angles
Acute, right, and obtuse angles are all angles that are classified by their degree measure. Acute angles have a degree measure less than 90$^{\circ}$. Right angles have a degree measure equal to 90$^{\circ}$. Obtuse angles have a degree measure greater than 90$^{\circ}$.

Interior and Exterior Angle Theorems:

Given two parallel lines (n and m) and their transversal (t) there are 8 angles that are created. (See the image$^6$ below)

This section aims to introduce and prove the theorems that are related to these 8 angles and their congruence. Use the image above to guide your understanding of the following theorems.

We will use some of these theorems in proofs involving triangles later on. (Note: All of these theorems require that n $\parallel$ m and the existence of their transversal t.)

Reference the image above for each of the proof below

Proof of Theorem 1: Assume n and m are parallel. Since $\angle{3}$ and $\angle{4}$ make up line m they are supplementary. This same relationship holds with $\angle{5}$ and $\angle{6}$. Call the point of intersection between lines m and t point A, and call the point of intersection between lines n and t point B. Now assume that m$\angle{3}\neq\angle{6}$. This means $\angle{3}<\angle{6}$ or $\angle{3}>\angle{6}$. Without loss of generality claim $\angle{3}>\angle{6}$. There there is some ray that starts at point A such that the ray passes through n at some point (call it D) and $\angle{DAB}\cong\angle{6}$. Now there is some point (call it E) on line n such that $\overline{BE}\cong\overline{DA}$. Therefore, by SAS congruence we have $\triangle{EBA}\cong\triangle{DBA}$. Therefore we have $\angle{5}\cong\angle{BAE}$. Finally since $\angle{5}$ and $\angle{6}$ are supplementary we have $\angle{BAE}\cong\angle{DAB}$ which would mean that D, A, and E are collinear. This is a contradiction since by design A is on line m and D and E are on line n. A similar contradiction will be obtained if we assume $\angle{6}>\angle{3}$. Therefore, by way of contradiction we have $\angle{3}\cong\angle{6}$ and $\angle{4}\cong\angle{5}$ QED

Proof of Theorem 2: We know that $\angle{3}\cong\angle{6}$ and $\angle{4}\cong\angle{5}$.We also know that $\angle{1}$ and $\angle{3}$ are supplementary, and $\angle{6}$ and $\angle{8}$ are supplementary. Therefore we know that since $\angle{3}\cong\angle{6}$ that $\angle{1}\cong\angle{8}$. This process can be repeated for each pair of alternate exterior angles. QED

Proof of Theorem 3: We know that $\angle{1}$ and $\angle{2}$ are supplementary. We also know that $\angle{1}$ and $\angle{3}$ are supplementary with a similar relationship for the other angles. Therefore, since $m\angle{1}+m\angle{3}=m\angle{1}+m\angle{2}=180$ we have that $\angle{2}\cong\angle{3}$. Now we also know $\angle{3}\cong\angle{6}$ by theorem 1, so we have $\angle{2}\cong\angle{6}$. A similar argument can be done for each pair of corresponding angles. QED

Proof of Theorem 4: By Theorem 3 we know that corresponding angles are congruent. Therefore, we have $\angle{1}\cong\angle{5}$. We know that $\angle{1}$ and $\angle{3}$ are supplementary so we know that $\angle{5}$ and $\angle{3}$ must also be supplementary. This proof is similar for $\angle{4}$ and $\angle{6}$. QED

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