NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Congruence and similarity algorithms

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In a triangle ABC, side AB = 6 cm, side BC = 8 cm, and angle B = 90°. In triangle DEF, side DE = 6 cm, side EF = 8 cm, and angle E = 90°.

Using the conditions for congruency of triangles, determine whether triangle ABC is congruent to triangle DEF. Provide your answer as a single word: 'Congruent' or 'Not congruent'.

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Worked solution

  1. Given triangle ABC with side AB = 6 cm, side BC = 8 cm, and angle B = 90°. This means triangle ABC is a right-angled triangle with the right angle at vertex B. The sides adjacent to the right angle are AB and BC, so the hypotenuse is AC.
  2. Given triangle DEF with side DE = 6 cm, side EF = 8 cm, and angle E = 90°. This means triangle DEF is also a right-angled triangle with the right angle at vertex E. The sides adjacent to the right angle are DE and EF, so the hypotenuse is DF.
  3. To determine if triangle ABC is congruent to triangle DEF, we apply the RHS (Right angle-Hypotenuse-Side) congruence rule. This rule states that if two right-angled triangles have their hypotenuses equal and one pair of corresponding legs equal, then the triangles are congruent.
  4. We compare the corresponding sides. In triangle ABC, AB = 6 cm and BC = 8 cm. In triangle DEF, DE = 6 cm and EF = 8 cm. Since AB corresponds to DE (both 6 cm) and BC corresponds to EF (both 8 cm), the two legs are equal.
  5. We now check the hypotenuses. Using the Pythagorean theorem: for triangle ABC, hypotenuse AC = sqrt(AB² + BC²) = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10 cm. For triangle DEF, hypotenuse DF = sqrt(DE² + EF²) = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10 cm. Therefore, the hypotenuses are equal.
  6. Since both triangles are right-angled, have equal hypotenuses (10 cm), and one pair of corresponding legs equal (6 cm and 8 cm), the RHS congruence condition is satisfied. Therefore, triangle ABC is congruent to triangle DEF.

Reference answer: Congruent

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