4.7 Geometrical reasoning

Learning objectives

4.7A Give informal reasons, where required, when arriving atgive informal reasons, where required, when arriving at numerical solutions to geometrical problems Reasons will only be required for geometrical calculations based on lines (including chords and tangents), triangles or polygons

A geometrical reason names the fact that makes a numerical step valid. Write the calculation and its reason together so each new angle can be checked.

Calculation fact	Accepted reason
total $180^\circ$	angles on a straight line / in a triangle
total $360^\circ$	angles around a point / in a quadrilateral
equal base angles	angles in an isosceles triangle
equal or supplementary circle angles	name the relevant chord, tangent or cyclic theorem

Mark known values, find one angle at a time, state the property used, then substitute that result into the next shape. Finish by checking every local angle sum.

A calculation such as $180-125=55$ is not itself a reason. State 'angles on a straight line sum to $180^\circ$ '.

Higher-tier reasoning should use a precise standard statement: vertically opposite angles are equal; alternate or corresponding angles are equal for parallel lines; opposite angles of a cyclic quadrilateral sum to $180^\circ$ .

Part of solution	What to write
value	the equation or angle calculation
relationship	equal, supplementary, parallel, tangent or same arc
justification	the named theorem with enough geometric context

If $ABCD$ is cyclic and angle $A=112^\circ$ , then angle $C=68^\circ$ because opposite angles in a cyclic quadrilateral sum to $180^\circ$ .

Do not write only 'circle theorem' or 'angles'. Name the exact theorem and ensure its conditions—such as cyclic vertices or parallel lines—are present.

Next Topic

—4.8 Trigonometry and Pythagoras’ theoremRead

4.7 Geometrical reasoning

Learning objectives

Give reasons in geometrical calculations

Use standard geometrical statements for angles

Next Topic