Geometrical Proof

Geometric proof involves using known rules about geometry to prove a new statement about geometry
A proof question might start with “Prove…” or “Show that …”
The rules that you might need to use to complete a proof include;
- Properties of 2D shapes
  - Especially triangles and quadrilaterals
- Basic angle properties
- Angles in polygons
- Angles in parallel lines
- Congruence and similarity
- Pythagoras theorem
You will need to be familiar with the vocabulary of the topics above, in order to fully answer many geometrical proof questions

Usually you will need to write down two or three steps to prove the statement
At each step, you should write down a fact and a reason
- For example, “AB = CD, opposite sides of a rectangle are equal length”
The proof is complete when you have written down all the steps clearly
- Use the diagram!
- Add key information such as angles or line lengths to the diagram as you work through the steps
  - but you must write them down in your written answer too!

Points or vertices of a shape are labelled with capital letters
- A, B, C and D are the vertices of the quadrilateral
- O is the centre of the circle
Two letters are used to represent the line between the points
- AB is the line between points A and B
Three letters are used to represent the angle formed by the three points
- Angle ABC is the angle between lines AB and BC
- The letter in the middle is the point where the angle is at
Multiple letters are used to represent the whole shape
- ABCD is a quadrilateral
- The letters are written down so that they go clockwise around the shape
If you use a variable to represent a length or an angle then write it down
- Angle ABC = $x$

Let a, b and c be the three interior angles in a triangle
Let d be the exterior angle next to the interior angle c
Split d into two angles by drawing a parallel line to the other side of the triangle
- There will be an angle alternate to angle a
- There will be an angle corresponding to angle b
Therefore the exterior angle is the sum of the two opposite interior angles

Exterior angle of a triangle

There are common phrases that are sufficient as explanations and should be learnt
- These will be what mark schemes look for
For triangles and quadrilaterals
- Angles in a triangle add up to 180°
- Base angles of an isosceles triangle are equal
- Angles in an equilateral triangle are equal
- Angles in a quadrilateral add up to 360°
- An exterior angle of a triangle is equal to the sum of the interior opposite angles
For straight lines
- Vertically opposite angles are equal
- Angles on a straight line add up to 180°
- Angles at a point add up to 360°
For parallel lines
- Alternate angles are equal
- Corresponding angles are equal
- Allied (or co-interior) angles add up to 180°
For polygons
- Exterior angles of a polygon add up to 360°
- The interior and exterior angle of any polygon add up to 180°

DO show all the key steps - if in doubt, include it
DON'T write in full sentences! For each step, just write down the fact, followed by the key mathematical reason that justifies it

In the diagram below, AC and DG are parallel lines. B lies on AC, E and F lie on DG and triangle BEF is isosceles.

4-5-3-geometrical-proof-we

Prove that angle EBF is $180 - 2 x$ . Give reasons for each stage of your working.

Mark on the diagram that triangle BEF is isosceles.

4-5-3-geometrical-proof-we-answer1

AC and DG are parallel lines, so using alternate angles we know that angle BEF = $x$ . Mark this on the diagram.

4-5-3-geometrical-proof-we-answer2

Write the fact, and the reason using the key mathematical vocabulary

angle BEF = $x$ , alternate angles are equal

Now using the fact that triangle BEF is isosceles, we can see that angle BFE = $x$ . Mark this on the diagram.

Write the fact, and the reason using the key mathematical vocabulary

angle BFE = $x$ , base angles of an isosceles triangle are equal

Now we can see that angle EBF is the last remaining angle in a triangle, and as the angles in a triangle sum to 180, angle EBF = $180 - 2 x$

Write the fact, and the reason using the key mathematical vocabulary

angle EBF = $180 - 2 x$ , angles in a triangle sum to 180

Geometrical Proof (Edexcel GCSE Maths: Foundation)