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New Zealand
Level 6 - NCEA Level 1

Congruent & Similar Triangles Revision


When we a working out circle geometry proofs, we aren't just limited to rules we have learnt in circle geometry. We can use all of a mathematic knowledge, including proofs that triangles are congruent or similar. Let's recap congruency and similarity so we're ready to use them in circle geometry.


Congruency Proofs

Here are the four proofs of congruency. You only need to use one of them to prove if any two triangles are congruent.

SSS- all sides are equal

AAS- two angles and one side are equal

SAS- two pairs of corresponding sides, and the pair of included angles are equal

RHS- right-angled triangles with equal hypotenuses and a pair of equal corresponding sides


Similarity Proofs

There are three distinct methods for proving two triangles are similar. You only need to use one of these proofs to show two triangles are similar, then all the features of similar triangles can be applied. 

AA- the triangles are equiangular (ie. all corresponding angles are equal)

SAS- two pairs of sides in the same ratio and equal included angles

SSS- three pairs of corresponding sides in the same ratio


Worked Examples

Question 1

Consider the following triangles. Using full geometric reasoning show that $\triangle ABC\equiv\triangle DEF$ABCDEF.

  1. In $\triangle ABC$ABC and $\triangle DEF$DEF:

Question 2

Consider the following triangles:

  1. Are the triangles similar, congruent or neither?






  2. What condition have you used to determine this?

    RHS: Two right-angled triangles with equal hypotenuses and one equal side


    AAS: Two equal angles and one included side


    SSS: Three equal sides


    SAS: Two equal sides and one included angle







Deduce and apply the angle properties related to circles


Apply geometric reasoning in solving problems

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