Question

1. a) What must be true in order to use the ASA Triangle Congruence Theorem to prove that two triangles are congruent? b) Two triangular pieces of artwork are hanging side by side. One is an equilateral triangle with a side length of 3ft. The second is an equilateral triangle with a side length of 4ft. Is it possible to prove the triangles are congruent using the ASA Triangle Congruence Theorem? Why or why not? c) Determine if the triangles are congruent. Explain your reasoning.

Answer 1

Using the ASA Triangle Congruence Theorem requires two angles and the included side of one triangle to be congruent to the corresponding parts of another triangle. The equilateral triangles with side lengths of 3ft and 4ft cannot be proven congruent through ASA since their sides are not equal. Thus, the triangles are not congruent.

Step-by-step explanation:

To use the ASA Triangle Congruence Theorem, two conditions must hold true: (1) two angles of one triangle must be congruent to two angles of the other triangle, and (2) the side between the two angles in one triangle must be congruent to the corresponding side between the two angles in the other triangle.

b) It is not possible to prove that the triangles are congruent using the ASA Congruence Theorem. The definition of an equilateral triangle guarantees that all angles are congruent; however, the triangles have sides of 3ft and 4ft, thus failing to meet the second requirement of the ASA theorem, which requires the corresponding sides to be congruent.

c) The two triangles are not congruent because for two triangles to be congruent, all corresponding sides and angles must match. In the case of the equilateral triangles with different side lengths, the corresponding sides do not match and therefore the triangles cannot be congruent.