Introduction

This lesson is the first section towards progressing past our findings we have discovered in the ``foundations'' section. This lesson will mainly be treated as a review, but will be keep as what is to come. It is important that we solidify our knowledge on what we know.

Thus far, we have discussed angle chasing, area formulas, and similar triangles. Throughout this chapter, we will continue using these ideas towards fresh new findings. In essence, throughout foundations, we learned about various techniques, but now we will apply them towards new ideas.

The ultimate goal is to build a repertoire where you understand each fact and can reason why each fact holds.

Handpicked Techniques

Philosophy

Two idea that I like to facilitate are

• "Make your diagram a work of art," and
• "Label everything you know."

In essence, your diagrams that you draw should be almost perfectly accurate and labeled. You want to be able to draw conclusions from your diagrams with ease.

Angle Chasing

In general, I like to say "When you don't know what to do, just angle chase!" You will be awfully surprised as to how well this goes. Unfortunately, it isn't so simple all the time. Here is a sample idea:

Identify any pairs of equal angles in the following setup. Consider a triangle $ABC$ and there exists a point $P$ such that $AP = BP = CP$.

Observations. The above is a simple example. Note that equal sides means equal angles. For instance, $\triangle APB$ is isosceles with $AP = BP$. It therefore follows that $\angle PAB = \angle PBA$. Likewise, $\angle PBC = \angle PCB$ and $\angle PCA = \angle PAC$.

In this instance, it is important to notice equal sides implies equal angles. Now, not every problem is like this. You may have to notice some different ideas to obtain angle relationships. But, it all revolves around the key idea of using angles to establish relationships which will ultimately lead to fresh conclusions.

Example 1 (American Mathematics Competition). Let $ABCD$ be a square and $E$ be a point outside square $ABCD$ such that $ABE$ is an equilateral triangle. Compute the measure of $\angle AED$, in degrees.

Consider triangle $AED$: note that as $AD = AB$ due to square $ABCD$ and $AB = AE$ due to equilateral triangle $ABE$. Hence, it follows that $AE = AD$ and $\triangle ADE$ is isosceles. Angle chasing,

It therefore follows that $\angle AED = \dfrac{180 - \angle AED}{2} = 15^{\circ}$.

Similar Triangles

It is very rare that there are problems that are solely angle chasing. But, when there are, you know what to do. In some cases, you may also have to establish relationships between lengths. We will look more into different relationships about lengths more throughout this chapter. But, for this section, we will touch generally on similar triangles. To recall, we know about the following similarity relationships on triangles:

• AA Similarity: Given two equal angles, then we may conclude the triangles are similar.
• SAS Similarity: Given two sides which come in equal proportions and congruent angles between them, then we may conclude the triangles are similar.
• SSS Similarity: Given three sides which come in identical proportions, then we may conclude the triangles are similar.

Recall, by noting similarity triangles, the powerful similarity ratio rises: for triangles $\triangle ABC \sim \triangle DEF$, then

Likewise, we have several congruency relationships:

• ASA Congruency: Given two equal angles and a congruent side between them, then we may conclude the triangles are congruent.
• SAS Congruency: Given two sides which are equal and congruent angles between them, then we may conclude the triangles are congruent.
• SSS Congruency: Given three sides which are identical, then we may conclude the triangles are congruent.
• AAS Congruency: Given two angles and a non-included corresponding side that are equal, the triangles are congruent.
• HL Congruency (for right triangles only): If the hypotenuse and one leg of a right triangle are equal to the corresponding hypotenuse and leg of another, the triangles are congruent.

Together with angle chasing, these techniques are extremely powerful as we will see in the sections to come. It is important to be careful and observant each time we encounter a problem. Being able to angle chase and incorporate similar triangles successfully requires careful analysis of the diagram in front of you.

Putting it all together

Example 2 (Folklore). Let $ABC$ be an isosceles triangle with $\angle BAC = 20^{\circ}$ and $AB = AC$. Point $E$ lies on segment $AB$ such that $AE = BC$. Find the measure of $\angle AEC$.

This is one of the most unique and challenging problems I have ever seen, but we surprisingly have enough tools for it. One would expect this to not really have a clean answer. After all, there is very little nice about triangles whose angle measurements are $20^{\circ}, 80^{\circ},80^{\circ}$. But, to one's surprise, there is plently nice about it. I will present the one I found while solving it along with my motivation, but I challenge you to find your own first!

The condition $AE = BC$ gives a neat clue. We will construct a triangle congruent to $ABC$ with base $AE$ instead of $BC$. Let $P$ be a point such that $\triangle PAE \cong \triangle ABC$. Nicely enough,

As $PA = AC$, then $\triangle PAC$ is equilateral. Then,

As $EP = PC$, then $\angle PEC = 70^{\circ}$. As $\angle PEA = 80^{\circ}$, then $\angle AEC = 150^{\circ}$.

Away from rigid, solid objects

As we begin a new chapter, here is a bit of a sneak peak of what we will be entering into. In general, we have usually talked about things like triangles, quadrilaterals, and such. But, we haven't talked too much about circles, and there is so much to talk about with them.

Throughout our discussions, you will realize that "you know very little about circles", ironically. And thus, when you have circles, make sure to connect intersections, centers of circles, and such. By creating more segments, you will be able to draw better conclusions as many, if not all, theorems that you know about tend to be about lengths!

Practice Problems