Question

Barbara and Sherwin want to use the SSS Triangle Congruence Theorem to see if their triangular slices of watermelon are congruent.

They each measure two sides of their slices. Barbara measures sides of lengths 7 inches and 6 inches, while Sherwin measures sides of lengths 8 inches and 5 inches.

Do they need to measure the third sides of their slices to determine whether they are congruent? If so, what must the side lengths be for the slices to be congruent? Explain.

Answer 1

No, they don't need to measure the third sides -- regardless of the measure of the third sides, we already have enough information to know that they cannot possibly be congruent.

Explanation:

Main concepts

1. Proving by way of contradiction

2. SSS triangle congruence

3. Proving these triangles can't be congruent

Concept 1. Proving by way of contradiction

To prove something by contradiction, we make a single assumption: usually that the opposite outcome is true.

Then, follow the logic from the given information and the contradiction assumption to see if a contradiction occurs. If so, since only one assumption was made, and the rest was based on facts, the contradiction assumption must be false.

Concept 2. SSS Triangle congruence

To use SSS Triangle congruence, each side of one triangle must correspond with each side of the other triangle, and each pair of corresponding sides must be congruent (the same length).

If even one pair of corresponding sides is not congruent, the triangles are not congruent. However, beware: occasionally, one needs to rotate or flip the triangles to ensure that the correct sides are corresponding.

Concept 3. Proving these triangles can't be congruent

Barbara's triangle has sides 7in and 6in.

Sherwin's triangle has sides 8in and 5in.

By way of contradiction, let's assume that the triangles are congruent. (this is our contradiction assumption).

Then each side of Barbara's triangle must correspond with a side of Sherwin's triangle and each pair of corresponding sides must be congruent.

Therefore, Barbara's 7in side, must correspond with a 7in side in Sherwin's triangle.

Since the two given sides in Sherwin's triangle are 8in and 5in, based on the assumption that the triangles are congruent, the last side of Sherwin's triangle must be 7in.

However, Barbara's triangle also contains a 6in side, which must also correspond with a 6in side in Sherwin's triangle. Since based on the assumption that the triangles are congruent, the third side of Sherwin's triangle was 7in, NONE of the sides in Sherwin's triangle can be 6in.

Effectively, the third side of Sherwin's triangle needs to be both 6in & 7 in simultaneously (which obviously isn't possible), which contradicts the assumption that the triangles were congruent. This means that the assumption that the triangles were congruent is incorrect.

Therefore, the triangles cannot be congruent.