Explain why knowing a combination of four pairs of equal sides or equal angles guarantees one of the congruence relationships.

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To understand why knowing a combination of four pairs of equal sides or equal angles guarantees one of the congruence relationships, we need to look at the concept of congruent triangles.

Congruent triangles are triangles that have the exact same shape and size. When two triangles are congruent, it means that all corresponding sides and angles are equal. Congruent triangles can be proven using different methods, and one of the most fundamental methods is the Side-Angle-Side (SAS) congruence criterion.

According to the SAS criterion, if we have two triangles with two pairs of congruent sides and one pair of congruent angles included between these sides, then the two triangles are congruent.

Now, let's apply this criterion to the given scenario of having a combination of four pairs of equal sides or equal angles.

First, let's consider the case of four pairs of equal sides. If we have two triangles with four pairs of equal sides, it means that all sides of each triangle are congruent. If this is the case, then we already have two pairs of congruent sides.

To satisfy the SAS criterion, we only need one pair of congruent angles included between these congruent sides. Since all angles in a triangle add up to 180 degrees, having four pairs of equal sides guarantees that at least one pair of angles is congruent. Therefore, by the SAS criterion, the two triangles are congruent.

Now, let's consider the case of four pairs of equal angles. If we have two triangles with four pairs of equal angles, it means that all angles of each triangle are congruent. If this is the case, then we already have two pairs of congruent angles.

Again, to satisfy the SAS criterion, we only need one pair of congruent sides corresponding to these congruent angles. Since all angles in a triangle are less than 180 degrees, having four pairs of equal angles guarantees that at least one pair of corresponding sides is congruent. Therefore, by the SAS criterion, the two triangles are congruent.

In summary, knowing a combination of four pairs of equal sides or equal angles guarantees one of the congruence relationships because it satisfies the SAS criterion, which states that two triangles are congruent if they have two pairs of congruent sides and one pair of congruent angles included between these sides.