Explain why knowing a combination of four pairs of equal sides or equal angles guarantees one of the congruence relationships.
To understand why knowing a combination of four pairs of equal sides or equal angles guarantees one of the congruence relationships, we need to recall the concept of congruence in geometry.
In geometry, two figures are said to be congruent if they have exactly the same shape and size. Congruence can be established using different criteria, such as the Side-Side-Side (SSS) criterion, the Side-Angle-Side (SAS) criterion, or the Angle-Side-Angle (ASA) criterion, among others.
Now, let's consider a scenario where we have two polygons and we know that their corresponding sides are equal. We can visualize this as follows:
Polygon A: AB = CD, BC = DE, CD = EF, and DA = FE
Polygon B: PQ = RS, QR = ST, RS = TU, and PR = UT
By knowing the combination of four pairs of equal sides in each polygon, we can assert that the corresponding angles between these sides are equal as well. This concept is known as the Angle-Side-Angle (ASA) criterion.
For example, in Polygon A, if we know that AB = CD, BC = DE, and DA = FE, then we can confidently state that ∠A = ∠C, ∠B = ∠D, and ∠D = ∠E. Similarly, in Polygon B, knowing that PQ = RS, QR = ST, and PR = UT would allow us to conclude that ∠P = ∠R, ∠Q = ∠S, and ∠S = ∠T.
By establishing the equality of corresponding angles, we can guarantee that the two polygons have the same shape. This, combined with the fact that we already know they have the same size because of the equal sides, allows us to say that the two polygons are congruent.
In summary, when we have a combination of four pairs of equal sides (or equal angles) in two polygons, we can rely on the Angle-Side-Angle (ASA) criterion to conclude that the two polygons are congruent.