Geometrical Proof (Edexcel IGCSE Maths)

Spec point — 4.7A

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 = $x$

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

Method 1 - Using angles in a triangle

• The interior angles in the triangle add up to 180°

  • a + b + c = 180°

• An interior angle and its exterior angle add up to 180°

  • c + d = 180°

• The right-hand sides of the equations are both equal (180°)

  • Therefore, the left-hand sides are equal

    • a + b + c = c + d

  • Subtract c from both sides

    • a + b = d

• Therefore the exterior angle is the sum of the two opposite interior angles

Method 2 - Using parallel lines

• 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°

Exam Hint

• 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

Worked Example 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.

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

Answer:

Mark on the diagram that triangle BEF is isosceles

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

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

Final answer: 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

Final answer: 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 = *$180-2x$

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

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