Formal geometric proof

Objective. Link congruence, similarity and circle theorems.

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Learn: the key idea

Formal geometric proof builds a logical chain linking known facts (congruence conditions, similarity ratios, circle theorems) to prove a new geometric statement step by step, each justified by a named reason. State which congruence condition (SSS, SAS, ASA, RHS) or circle theorem applies at each stage, and show how it leads to the next piece of information needed. Diagrams should be labelled clearly so each triangle or angle referenced in the proof is unambiguous. To prove two triangles sharing a side and equal base angles are congruent, cite ASA explicitly rather than just asserting congruence. Every claim in a formal proof needs its supporting reason stated.

Worked example

In triangle ABC, AB = AC and the angle bisector from A meets BC at D. Prove that triangles ABD and ACD are congruent.

1 $AB = AC is given$
2 $Angle BAD = angle CAD since AD bisects angle A$
3 $AD is common to both triangles$
4 $By SAS, triangles ABD and ACD are congruent$

Answer: Triangles ABD and ACD are congruent by SAS

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Common mistake. Students often state 'triangles are congruent' without naming the specific condition (SSS, SAS, ASA, RHS) used, which loses marks since the logical justification itself is what is being assessed.

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