Two Triangles with Five Pairs of Congruent Parts

To create two triangles that have five pairs of congruent parts but are not congruent, we need to consider the different criteria for triangle congruence.

1. Side-Side-Side (SSS) Criterion:
If two triangles have all three sides congruent, they are congruent. Let's assume Triangle ABC and Triangle XYZ both have sides AB = XY, BC = YZ, and AC = XZ.

2. Side-Angle-Side (SAS) Criterion:
If two triangles have two sides and the included angle congruent, they are congruent. Let's assume Triangle ABC and Triangle XYZ both have sides AB = XY, angle BAC = angle YXZ, and side AC ≠ side XZ.

3. Angle-Side-Angle (ASA) Criterion:
If two triangles have two angles and the included side congruent, they are congruent. Let's assume Triangle ABC and Triangle XYZ both have angle BAC = angle YXZ, angle ABC = angle XYZ, and side AC ≠ side XZ.

4. Angle-Angle-Side (AAS) Criterion:
If two triangles have two angles and a non-included side congruent, they are congruent. Let's assume Triangle ABC and Triangle XYZ both have angle BAC = angle YXZ, angle ABC = angle XYZ, and side BC ≠ side YZ.

Creating the Triangles:
To create two triangles with five pairs of congruent parts but are not congruent, we can use the SAS criterion.

Triangle ABC:
- Side AB = Side XY
- Angle BAC = Angle YXZ
- Side AC ≠ Side XZ

Triangle XYZ:
- Side XY = Side AB
- Angle YXZ = Angle BAC
- Side XZ ≠ Side AC

If we draw these triangles, we can see that they have five pairs of congruent parts:
1. Side AB = Side XY
2. Angle BAC = Angle YXZ
3. Side AC ≠ Side XZ
4. Angle ABC = Angle XYZ
5. Side BC ≠ Side YZ

However, the triangles are not congruent because the third side, AC and XZ, are not equal. This violates the SAS criterion for triangle congruence. Hence, even though the triangles have five pairs of congruent parts, they are not congruent.

Overall, by understanding the different criteria for triangle congruence and strategically selecting the lengths of the sides and angles, we can create two triangles with five pairs of congruent parts while ensuring they are not congruent.