Geometric Proof (AQA GCSE Further Maths)

What is a geometric 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;
  • Basic angle facts and properties of 2D shapes
    • Examples
      • "the angle sum in any quadrilateral is 360°"
      • "an isosceles triangle has two equal sides/two equal angles"
      • "vertically opposite angles are equal"
  • Angles involved with parallel lines
    • alternate and corresponding angles
  • Circle theorems
    • there are a few of these!
    • these are dealt with separately in the revision note Algebraic Circle Theorems

• • Pythagoras' theorem (only applies to right-angled triangles)
    •  where  is the hypotenuse
  • Congruence and similarity
    • congruent shapes are identical
    • similar shapes have equal angles; side lengths are in the same ratio to each other
• You will need to be familiar with the vocabulary of the topics above
  • reasons are often required in order to fully explain a geometrical proof

How do I write a geometric proof?

• Usually you will need to write down a few steps to prove the statement
  • At any step where an angle fact is used you should write down the fact and a reason
    • do this even for the most basic facts
      • e.g.  angles on a straight line add up to 180°
• The proof is complete when you have written down all the steps clearly
• Before getting into your steps of working for your proof
  • underlining key words or facts in the question
  • mark any important information given in words to a given diagram
  • add any information (such as line lengths and/or angles) to the diagram using facts you know
    • you can write down the steps and reasons for these later

What notation should I use?

• Labelling vertices (corners) and lengths are done in capital letters
  • Examples
    • "from A to B ..."
    • "side lengths AB and BC are equal..."
    • "triangle ABC is ..."
• Labelling angles can be done by writing "angle ABC"
  • you may see other ways including
    • single letters ("angle B")
    • "hat notation"
      
    • "∠ABC" which would be read as "angle ABC"
• Of course if an angle is marked with a letter (x or θ are common) use that
  • If not, you can add your own letters in, but do make them clear on the diagram

What are algebraic circle theorems?

• This part of geometric proof focuses on using circle theorems
  • Here's a reminder of them
    • the top-right diagram is also called the "Alternate Segment Theorem"

• Rather than finding the values of missing angles you may have to
  
  • prove/show a result that links two or more unknowns (letters)
  • e.g.  show that ""
    • there is no requirement to find the actual values of  and  separately
    • just that they sum to 90

How do I prove or show a result using circle theorems?

• Other skills beyond circle theorems may be involved
  • • angles in basic shapes, parallel lines, congruence, Pythagoras' Theorem, etc
• Have in the back of your mind the result you are aiming for
  • This will help stop you from trying to find values for particular unknowns
    • which may be impossible
• Use angle facts alongside circle theorems to find a relationship between the the unknowns
  • e.g. if the three angles in a triangle are 65°, and , then the fact is and the reason is that angles in a triangle sum to 180°
• You may have to use some basic algebra skills to get the required result
  • simplifying, factorising, expanding, ...
    • e.g. if the two non-right angle angles in a triangle in a semi circle are  and  then
      
      but we can then simplify by dividing through by 5 to give
      
• You may be given extra information in later parts of a question that do allow the unknowns to be found
  • e.g.  Having shown , if we are then told that  then we can find both  and 
    • using simultaneous equations