# Geometric Proof

# Introduction to Geometric Proof

- A
**geometric proof**is a method of establishing the truth of a geometric proposition by using a set of established rules based on previously accepted or proven propositions. - Geometric proofs utilise
**axioms**or**postulates**, which are statements accepted as true without requiring further proof, and**theorems**, which are propositions that need to be proven on the basis of established postulates.

# Components of a Geometric Proof

- Each geometric proof starts with a
**given**statement which is the known information. - The proof method requires identifying the
**goal**statement you need to prove. - You construct a
**logical argument**to link the given statement to the goal statement. - The argument consists of a series of steps, each of which must be justified by a postulate, axiom, definition, or previously proven theorem.

# Methods of Geometric Proof

- A
**two-column proof**is the most classical form of proof. You write the steps of the proof in the left column and the reasons for taking each step in the right column. - A
**paragraph proof**is written in paragraph form. It can be less structured than a two-column proof but must contain all the steps needed to move from the given statement to the goal statement. **Informal proofs**are a loose construction of logic or sample drawing. These proofs lack the structure of the two-column and paragraph methods but are a good starting point when first learning about geometric proofs.

# Working Through a Geometric Proof

- Begin with the
**given statement**and consider the**goal statement**you need to prove. - Identify what theorems and properties you can use to create a series of linked statements starting with the given statement and ending with the goal statement.
- Write each step, making sure to justify each one. Double-check that each step follows logically from the one before it.
- The
**final step**of your proof should be the goal statement.

# Essential Concepts for Geometric Proof

- Familiarize yourself with core geometry concepts, including parallel and perpendicular lines, triangles, circles, and polygons.
- Thoroughly understand the properties of geometric shapes, as these are frequently used to justify steps in geometric proofs.
- Be comfortable working with angles and practise using definitions and theorems related to angles.

# Tips for Successful Geometric Proof

**Practice**regularly to understand the logical flow of mathematical arguments.- Draw a
**diagram**if one is not provided. Visual representation can greatly aid understanding. - If the path to the goal is not immediately apparent, try considering
**intermediate goals**. - Write clearly and concisely, but be sure every
**step is justified**. - Remember, mastering geometric proofs requires time and persistence but it will significantly enhance your
**analytical thinking**skills.