New Zealand

Level 6 - NCEA Level 1

Lesson

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.

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

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

Consider the following triangles. Using full geometric reasoning show that $\triangle ABC\equiv\triangle DEF$△`A``B``C`≡△`D``E``F`.

In $\triangle ABC$△

`A``B``C`and $\triangle DEF$△`D``E``F`:

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

Deduce and apply the angle properties related to circles

Apply geometric reasoning in solving problems