Question

Really need help Match each set of conditions with the corresponding relationship between ABC AND XYZ and the criterion(if any) that proves the relationship.

Answer 1

Answer:

As mentioned in the steps of explanation.

Step-by-step explanation:

(1). ΔABC and ΔXYZ are congruent by the SSS criterion →   AB=XY, BC= YZ and CA=ZX, criterion that prove this relation as,

SSS criterion or postulate -if three sides of one triangle is same (congruence) to the corresponding three sides of the another triangle then the two triangle are congruence.

(2). ΔABC and ΔXYZ are congruent by the SAS criterion →  AB=XY, BC=YZ and angle B is congruent to angle Y, criterion that prove this relation as,

SAS criterion or postulate -if two sides and a angle between them in one triangle are congruence to the corresponding two sides and an between them in another triangle, triangle then the two triangle are congruence.

(3).   ΔABC and ΔXYZ are congruent by the ASA criterion →  AB=XY, and angles A and B are congruent to angles X and Y, criterion that prove this relation as,

ASA criterion or postulate - if two angles and one side between them in one triangle are congruent to the corresponding two angles and one side between them in another triangle,, triangle then the two triangle are congruence.

(4). ΔABC and ΔXYZ are not necessarily congruent →  Angles A, B and C are congruent to angles X, Y and Z, respectively, postulate that prove this relation as,

AAA similarity postulate - if three angles of a triangle is congruent (same) to the corresponding three angles of the another triangle. The two triangle must be similar but it may be congruent (not necessary).

Answer 2

The triangle similarity postulates are:

SSS: side-side-side - if all three sides of one triangle is congruent to all three sides of another triangle, then the two triangles are congruent.

SAS: side-angle-side - if one triangle's two sides and the angle between them are congruent to another triangle's two sides and the angle between them, then the two triangles are congruent.

ASA: angle-side-angle - if one triangle's two angles and the side between them is congruent to another triangle's two angles and the side between them, then the two triangles are congruent.

I recommend drawing the triangles and drawing the dash for congruent sides and marking congruent angles.

1) For ABC and XYZ to be congruent by SSS,
Side AB has to equal XY,
Side BC has to equal YZ,
Side CA has to equal ZY.

2) For ABC and XYZ to be congruent by SAS,
Two sides of triangle ABC have to be the same length as two sides of triangle XYZ, and the angle between the sides of both triangles have to be congruent.

3) For ABC and XYZ to be congruent by ASA,
Two angles of triangle ABC have to be the same as two angles of XYZ, and the two sides that create those angles of ABC has to be equal to the two adjacent sides of triangle XYZ.

4) ABC and XYZ are not necessarily congruent when corresponding sides don't meet the SSS, SAS, ASA (or AAS, derived from ASA).
If only angles are congruent and no congruent sides are mentioned, then the triangles are not necessarily congruent.