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If you’re here, you’ve learned which tools you have to work with, and the jobs they do. This means you’re ready to start putting your new way of thinking to work and apply it to navigating the intricacies of more complicated geometric relationships. Get ready to flex your mental muscles and dive into the world lines,  planes, and angles!  All of my study guides are downloadable so you can keep them handy to use as a reference while you're working problems.  If at any point you feel the need for guidance or if things seem overwhelming, reach out. Consultations are not about finding someone to hold your hand; they're about customizing your learning experience and ensuring that you thrive independently.

One of the initial challenges you'll encounter is identifying whether a set of lines is parallel or perpendicular. Fortunately, you've got the tools to crack this geometric code. Let's break it down.

Two lines are parallel if they never meet, no matter how far you extend them. If they never intersect, they must have the same slope. In other words, they maintain a consistent distance from each other into infinity. To solidify your understanding,  picture real-world examples where parallel lines are prevalent. Think railroad tracks or the yard lines on a football field.

On the other hand, perpendicular lines intersect at a right angle, forming a perfect 90-degree angle between them. Picture the corners of a square or the cross-section of a neatly folded piece of paper. Identifying perpendicular lines involves recognizing this distinct relationship, and the tools you've acquired will help you decipher these geometric puzzles.

Now that you've got a sense of how things build in one-dimension. Let's take a step further and look at planes which are are two-dimensional.

Just as lines can be parallel, planes can also share this relationship. Imagine slicing through space with a plane—parallel planes maintain the same separation as they extend infinitely. if you're sitting in your room studying right now, look up and then look down.  The ceiling and the floor each representing a parallel plane. 

Again, you can apply the one dimensional rules to two dimensions here.  Perpendicular planes are two large, flat surfaces that intersect each other at a right angle, creating a perfect 90-degree angle where they meet. Still sitting in your room looking at the ceiling and the floor?  Now look at the place where the floor meet a wall.  The floor and the wall are two perpendicular planes.

And then the angles ... oh, the angles ... Why are they such a big deal anyway? There are a lot of them that you need to learn to recognize, construct, and use to solve problems. It turns out that if pairs of them are congruent, meaning they have the same measure, then the lines being intersected are parallel. This relationship provides a powerful tool for establishing parallelism and deepens your understanding of geometric configurations.  When it comes to things like this, seeing is believing.  Drawings help, so I'll leave you to browse the study guide for illustrated examples of each of them, but for starters let's talk about just one.

Alternate exterior angles are formed when a transversal intersects two parallel lines. Picture a set of parallel railroad tracks, and a third track intersecting them at an angle. The angles on the outer sides of this transversal and on opposite sides of the train tracks are alternate exterior angles.

Congruent angles and parallel lines are also your first exposure to "if...then" statements which can also be called Theorems, and most theorems have a converse which is also true.  Brace yourself!  This means you're ready to start venturing into the land of proofs. 

Proofs are the backbone of geometry—they're the means by which you validate your arguments and establish the certainty of geometric truths. For example if two lines are parallel and intersected by a transversal, then corresponding angles are congruent. The converse of this would be that if corresponding angles are congruent, then the lines intersected by the transversal that formed the angles are parallel.

I've said it before and I'll say it again: The key to success in geometry is learning to articulate your reasoning clearly and persuasively, just like a skilled debater presenting a case in court. Each step in a proof is a logical progression, building upon the previous one to construct an airtight argument. Think of it as building a set of statements where each contributes an overarching  geometric truth.


Parallel Lines:

Perpendicular Lines:

Parallel Planes: 

Perpendicular Planes: 

Alternate Exterior Angles: 

Geometric Proofs:

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