Question

Which criteria for triangle congruence can be used to prove the pair of triangles congruent? explain?

please help :)

Answer 1

all except AAA

Explanation:

assuming that the shape above is a rectangle, all criteria for congruence can be proven.

SSS:

• sides marked once are equal (S)
• sides marked twice are equal (S)
• diagonal is a shared side (S)

SAS:

• sides marked once are equal (S)
• all vertices of a rectangle are `90^@` (A)
• sides marked twice are equal (S)

AAS/ASA/SAA:

• all vertices of the rectangle are `90^@` (A)
• sides marked once are equal (S)
• angles on the top-right corner of one triangle, and on the bottom-left corner of the other, are alternate
• alternate angles are equal (A)

while it is true that both triangles have equal angles, AAA is not sufficient proof for congruence - only for similarity,

all angles are equal between similar triangles, and so all of their sides are in the same ratio.

however, one triangle can have larger sides than another. this is true for any ratio of sides that is not `1:1`.

e.g.

here, the ratio between sides is `1:3`.

similar triangles have equal angles, and all of their sides are in the same ratio to each other.

congruent triangles have equal angles, and all of their sides have the ratio of `1:1` to each other.

all congruent triangles are similar, but not all similar triangles are congruent.

therefore SSS, SAS or ASA is enough to prove both congruence and similarity, but AAA on its own is enough to prove similarity but not congruence.