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MathidiaMathematics Academy
Proof, modelling and Year 10 synthesis
Lesson 72 of 75
Easy practice

Geometric proof

Objective. Build linked angle and congruence arguments.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Geometric proof links known angle facts and congruence conditions into a logical chain to justify a conclusion about a shape. Key facts include angles on a straight line sum to 180°, angles in a triangle sum to 180°, and congruence tests SSS, SAS, ASA, RHS confirm two triangles are identical in shape and size. To build a proof, state each fact used and why it applies, moving step by step from given information to the required conclusion. For example, to prove two triangles are congruent, identify three matching pairs of sides or angles satisfying one condition, then state the congruence rule explicitly, such as SAS when two sides and the included angle match.

Worked example

Triangles ABC and DEF have AB = DE, BC = EF, and angle B = angle E. Give a geometric proof that the triangles are congruent.

Diagram of a triangle
  1. 1Identify matching sides: AB = DE and BC = EF are given
  2. 2Identify the included angle: angle B = angle E is given, and it lies between the matching sides
  3. 3Two sides and the included angle match, satisfying SAS
  4. 4By SAS, triangle ABC is congruent to triangle DEF

Answer: Congruent by SAS

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Common mistake. A common error is asserting triangles are congruent without naming a valid test or matching corresponding parts correctly, e.g. claiming SSA proves congruence, which is not a valid congruence condition in general.

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