Geometry is a fascinating branch of mathematics that deals with the shapes, sizes, and properties of figures and spaces. One of the key aspects of studying geometry is understanding geometric proofs.
Unlike other areas of mathematics, geometry often requires you to work backward: you’re given a conclusion, and your task is to justify it. This is where geometric proofs come in.
In this guide, we’ll start with the basics and work our way up to applications of proofs beyond the world of geometry. Read it straight through for a comprehensive overview, or jump around to find the piece of insight you need.
Regardless of where you are at in your learning journey, this guide has a little something for everyone.
Table of Contents
Core Foundations of Geometry
1. The Power of Deductive Reasoning
Before we dive in it’s important to understand the power of deductive reasoning in geometry.
Deductive reasoning is the process of reaching a logically certain conclusion from given premises. In geometry, we use deductive reasoning to derive conclusions from definitions, postulates, and previously proven theorems.
This is the foundation of geometric proofs.
2. The Role of Definitions, Postulates, and Theorems
In geometric proofs, we often rely on…
- Definitions (precise descriptions of geometric concepts),
- Postulates (accepted truths without proof), and
- Theorems (statements proven based on definitions, postulates, and previously proven theorems)
Understanding these elements and how they interact is crucial for constructing geometric proofs.
For example, the definition of a right angle (an angle measuring 90 degrees), the postulate that the angles in a triangle add up to 180 degrees, and the theorem that the base angles of an isosceles triangle are congruent are all essential tools for geometric proofs.
Four Styles of Geometric Proofs
At its core, a geometric proof is simply deductive reasoning that leads us to a concrete conclusion. But how do we show our reasoning? It turns out that we have some options.
In geometry, there are four main styles of proofs…
- Two Column Proofs
- Paragraph Proofs
- Flowchart Proofs
- Visual Proofs
These four are stylistically different but functionally the same, taking us from point A to point B with a series of logical steps.
Let’s start by exploring a single proof, showing that diagonals of a parallelogram bisect each other, using the first three styles.
Proving That Diagonals Of A Parallelogram Bisect Each Other
1. The Two Column Proof
ABCD is a parallelogram
AD ≅ BC and AB ≅ DC
Opposite sides of a parallelogram are congruent
∆ADB ≅ ∆BCA
Side-Side-Side (SSS) Congruence Postulate
AD bisects BC and BC bisects AD
Corresponding parts of congruent triangles are congruent (CPCTC)
2. The Paragraph Proof
Given that ABCD is a parallelogram, we know that AD ≅ BC and AB ≅ DC because the opposite sides of a parallelogram are congruent. By the Side-Side-Side (SSS) Congruence Postulate, we can conclude that ∆ADB ≅ ∆BCA. Therefore, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, AD bisects BC and BC bisects AD.
3. The Flowchart Proof
Proving the Pythagorean Theorem
Arguably the most famous theorem in geometry is the Pythagorean theorem. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In other words, a2 + b2 = c2
This theorem is a cornerstone of geometry and has a multitude of applications, from architecture to astronomy. It is also a product of the humble geometric proof.
In fact, there are several different geometric proofs for the pythagorean theorem, all using different deductive reasoning to arrive at the same conclusion. Let’s look one of the simplest and most elegant proofs of the bunch.
4. The Visual Proof
This visual proof of the pythagorean theorem shows how we can rearrange 4 identical right triangles within a square to leave us with either or c2 or a2 + b2.
Because the area of the square and the 4 right triangles is the same in both variations, the left over area represented by the white space must also be the same. Therefore, a2 + b2 must be equal to c2.
Direct vs. Indirect Proofs
The above examples take us from Point A to Point B using a series of direct logical steps. But it is also possible to prove some things indirectly. In an indirect proof, instead of proving a statement directly, you assume that the statement is false and then show that this assumption leads to a contradiction. This proves that the original statement must be true.
Let’s consider an example. Suppose we want to prove that if a line is drawn from the center of a circle to the midpoint of a chord, then the line is perpendicular to the chord.
In an indirect proof, we would start by assuming the opposite: that the line is not perpendicular to the chord. This means that there is another line from the center of the circle to the chord that forms a right angle. But this leads to a contradiction, because it would mean that there are two different lines from the center of the circle to the midpoint of the chord, which is impossible. Therefore, our original statement must be true.
Indirect proofs can be a bit trickier to understand than direct proofs, but they’re a powerful tool for solving problems that might be difficult to approach directly.
Taking Proofs Beyond the Geometry Classroom
While geometric proofs are a fundamental part of studying geometry, they’re also a fundamental aspect of mathematics as a whole. They provide a rigorous way to establish the truth and build upon existing knowledge. By constructing a proof, you’re not just showing that a statement is true—you’re showing why. This deepens your understanding and allows you to apply the same principles to other problems.
Whether you’re studying algebra, calculus, or number theory, the ability to construct a rigorous proof is a crucial skill. It allows you to establish truth, build on existing knowledge, and develop a deep understanding of mathematical principles.
Real-World Applications of Geometric Proofs
Furthermore, geometric proofs have many real-world applications. Engineers use geometric proofs to design and optimize structures. Computer scientists use them in graphics and algorithms. Physicists use them to understand the nature of the universe.
For example, consider the task of designing a bridge. Engineers need to ensure that the bridge is strong enough to support a certain weight. They might use geometric proofs to show that the forces on the bridge are distributed in a way that prevents it from collapsing.
Or consider the field of computer graphics. When rendering a 3D scene, a computer needs to calculate how light interacts with different surfaces. Geometric proofs can be used to show that the calculations are correct and that the rendered image accurately represents the scene.
These are just a few examples of how geometric proofs are used in the real world. They’re not just abstract exercises—they’re a way of ensuring that our buildings, our technology, and our understanding of the universe are sound.
The Beauty of Geometry
Geometry is not just about numbers and shapes. It’s a way of seeing the world. From the symmetry of a snowflake to the layout of a city, geometric principles are everywhere if you know where to look. Understanding geometric proofs allows you to appreciate this beauty on a deeper level.
How Emergent Education Can Help You Master the Geometric Proof
Mastering geometric proofs is a journey, but it’s a rewarding one. With each proof you construct, you’re not just solving a problem—you’re honing your logical reasoning skills, deepening your understanding of geometry, and participating in the grand tradition of mathematical proof.
At Emergent Education, we’re here to support you every step of the way. Whether you’re just starting with geometric proofs or looking to tackle more complex problems, our experienced tutors are ready to help.