Question

which of the following triangles are congruent by the sas congruence theorem? enter the lesser number first.
triangle #1: a triangle with an angle measuring 60 degrees between sides measuring 3 cm and 6 cm
triangle #2:
triangle #3:

Explanation:

Step1: Recall SAS Congruence Theorem

The SAS (Side - Angle - Side) Congruence Theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze Triangle #1, Triangle #2 and Triangle #3

Triangle #1: has sides \(3\mathrm{cm}\), \(6\mathrm{cm}\) and included angle \(60^{\circ}\).
Triangle #2: has sides \(3\mathrm{cm}\), \(6\mathrm{cm}\) and included angle \(60^{\circ}\).
Triangle #3: has sides \(3\mathrm{cm}\), \(6\mathrm{cm}\) and included angle \(60^{\circ}\).

Since all three triangles (Triangle #1, Triangle #2 and Triangle #3) have two sides and the included angle equal. But if we assume Triangle #1 is the description (text - based) and Triangle #2 and Triangle #3 are the visual ones. If we consider the problem as comparing the text - based triangle (Triangle #1) with the visual ones (Triangle #2 and Triangle #3), we note that Triangle #2 and Triangle #1 (by description) and Triangle #3 and Triangle #1 (by description) satisfy the SAS criterion. But if we assume that the problem is among the given figures (assuming Triangle #2 and Triangle #3 are the main candidates for comparison, since Triangle #1 is a text - based description which is same as the visual ones).

Answer:

\(1\) and \(2\), \(1\) and \(3\) (If we assume the text - based Triangle #1 is to be compared with the visual Triangle #2 and Triangle #3). If we assume that the problem is between the visual figures (Triangle #2 and Triangle #3 are congruent as they have \(3\mathrm{cm}-60^{\circ}-6\mathrm{cm}\) sides - angle - sides), the answer is \(2\) and \(3\)