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 $\triangle ABC\equiv\triangle DEF$△ABC≡△DEF.
In $\triangle ABC$△ABC and $\triangle DEF$△DEF:
Question 2
Consider the following triangles:
Are the triangles similar, congruent or neither?
Congruent
ANeither
BSimilar
CWhat condition have you used to determine this?
RHS: Two right-angled triangles with equal hypotenuses and one equal side
AAAS: Two equal angles and one included side
BSSS: Three equal sides
CSAS: Two equal sides and one included angle
D