Question

consider the two triangles shown below.
are the two triangles congruent?
choose 1 answer:
a yes
b no
c there is not enough information to say.

Explanation:

Step1: Identify congruence criteria

To determine if two triangles are congruent, we can use criteria like ASA (Angle - Side - Angle), SAS (Side - Angle - Side), SSS (Side - Side - Side), etc. Let's analyze the given triangles.

Step2: Analyze the given angles and side

In both triangles, we have a side of length 6. We also have two angles: \(61^{\circ}\) and \(96^{\circ}\). Let's check the angle - side - angle relationship. For the first triangle (let's say Triangle 1) and the second triangle (Triangle 2), the side of length 6 is between the \(61^{\circ}\) angle and the \(96^{\circ}\) angle in both triangles? Wait, no, actually, in triangle congruence, if we have two angles and the included side equal, then the triangles are congruent by ASA. Wait, let's calculate the third angle to confirm. The sum of angles in a triangle is \(180^{\circ}\). For a triangle with angles \(61^{\circ}\) and \(96^{\circ}\), the third angle \(x=180-(61 + 96)=180 - 157 = 23^{\circ}\). So both triangles have angles \(61^{\circ}\), \(96^{\circ}\) and \(23^{\circ}\), and they have a side of length 6. Now, looking at the diagram, the side of length 6 is between the \(61^{\circ}\) and \(96^{\circ}\) angles in both triangles? Wait, actually, in the first triangle (the left one), the side of length 6 is between the \(61^{\circ}\) and \(96^{\circ}\) angles? Wait, no, maybe the side of length 6 is the side that is between the two angles. Wait, actually, if we consider the ASA (Angle - Side - Angle) criterion: if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent. Here, both triangles have a side of length 6, and the two angles adjacent to this side ( \(61^{\circ}\) and \(96^{\circ}\)) are equal in both triangles. So by ASA congruence criterion, the two triangles are congruent.

Answer:

A. Yes