Geometrical-Proof Aqa Foundation

Geometrical proof

What is a 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

How do I write a geometrical proof?

• 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

What geometric notation should I use?

• 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 = 

How can I prove that the exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices?

• 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

What are common geometric reasons I can use?

• 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°