Explain Why Triangles with 8cm Side, 45° and 80° Angles May Not Be Congruent

Question

Alex and Heather each draw a triangle with one side of 8cm, one angle of 45° and one angle of 80°. Alex says that their triangles are congruent. Explain why Alex might not be correct. (2 marks)

Accepted Answer

Alex might be incorrect because the 8cm side may not be the included side between the 45° and 80° angles for both triangles. If one triangle has the 8cm side as the included side (satisfying ASA congruence) and the other has the 8cm side opposite one of the angles, the triangles will not be congruent.

Answer

Step 1: Recall the ASA (Angle-Side-Angle) congruence rule: For two triangles to be congruent via ASA, the side must be the included side (the side between the two given angles) in both triangles, with the corresponding angles and included side being equal. Step 2: Analyze the given information: Both triangles have two angles (45° and 80°) and one side (8cm), but the position of the 8cm side is not specified. Step 3: Consider two scenarios: 1) In one triangle, the 8cm side is the included side between the 45° and 80° angles; 2) In the other triangle, the 8cm side is opposite either the 45° angle or the 80° angle. Step 4: Conclude: Since the 8cm side may not be the corresponding included side for both triangles, the ASA congruence test is not satisfied, so the triangles do not have to be congruent.