Geometrical Proof

What Is Geometrical Proof?

A geometrical proof is a logical explanation that uses known geometric rules to show why a mathematical statement is true.

You might see phrases like:

• "Prove that..."
• "Show why..."

To build your proof, you'll need knowledge of:

• Angle rules (lines, triangles, polygons)
• Properties of 2D shapes (especially triangles and quadrilaterals)
• Congruence and similarity
• Pythagoras' Theorem
• Parallel line angle facts

 

How Do I Structure a Proof?

Follow this step-by-step method:

1. Mark known facts and key points on the diagram.
2. State what you know, and justify it with a reason.
3. Build up your logic step-by-step to reach the final result.
4. Use clear geometric notation.

Example Notation:

• AB: line between points A and B
• $\angle ABC$: angle at point B between lines AB and BC
• ABCD: quadrilateral labelled clockwise
• x: used for unknown angles or side lengths

 

Useful Geometric Reasons

Here are common reasons to use in proofs:

Triangles & Quadrilaterals

• Angles in a triangle add up to $180\degree$
• Base angles of an isosceles triangle are equal
• Angles in a quadrilateral add up to $360\degree$
• Exterior angle = sum of interior opposite angles

Straight Lines & Points

• Angles on a straight line add to $180\degree$
• Vertically opposite angles are equal
• Angles at a point add up to $360\degree$

Parallel Lines

• Alternate angles are equal
• Corresponding angles are equal
• Co-interior (allied) angles add up to $180\degree$

 

Example

Given: AC and DG are parallel lines. Triangle BEF is isosceles with BE = BF

 

Prove that $\angle EBF = 180 - 2x$. You must give reasons for each stage of working.

 

Step 1: Use Alternate Angles

Since AC \parallel DG:

• $\angle BEF = x$ (alternate angles)

Reason: Alternate angles are equal

Step 2: Use Isosceles Triangle Properties

BE = BF, so:

• $\angle BFE = x$

Reason: Base angles of an isosceles triangle are equal

Step 3: Use Triangle Angle Sum

• Triangle BEF has three angles: $\angle BEF, \angle BFE, \angle EBF$
• $\angle EBF = 180 - (x + x) = 180 - 2x$

Reason: Angles in a triangle add up to $180\degree$