Question

example 6
ex6: which congruency theorem can be used to prove that these triangles are congruent?
aas sss sas asa hl
rewatch

Explanation:

Step1: Analyze given triangle parts

From the diagram, \( VU = VT \) (marked with ticks), \( \angle VUS=\angle VTS \) (marked angles), and \( VS \) is common to both \( \triangle VUS \) and \( \triangle VTS \).

Step2: Match with congruence theorems

• SAS (Side - Angle - Side) requires two sides and the included angle. Here, we have two sides (\( VU = VT \), \( VS = VS \)) and the included angle (\( \angle VUS=\angle VTS \))? Wait, no, actually, the included angle between \( VU \) and \( VS \) and \( VT \) and \( VS \): Wait, the angles at \( V \) are marked as equal? Wait, looking again, the segments \( VU \) and \( VT \) are congruent, \( VS \) is common, and the angle between \( VU \) and \( VS \) and \( VT \) and \( VS \) (the included angle) is equal? Wait, actually, the correct theorem here: Let's re - examine. The triangles \( \triangle VUS \) and \( \triangle VTS \): \( VU = VT \) (side), \( \angle VUS=\angle VTS \) (angle), and \( VS = VS \) (side). Wait, no, SAS is side - angle - side where the angle is between the two sides. So if \( VU = VT \), \( \angle UV S=\angle TV S \) (wait, maybe the diagram has \( \angle UV S\) and \( \angle TV S\) equal? Wait, the original diagram: the two triangles share \( VS \), \( VU = VT \), and the angle between \( VU \) and \( VS \) and \( VT \) and \( VS \) is equal. So that's two sides (\( VU = VT \), \( VS = VS \)) and the included angle (\( \angle UV S=\angle TV S \))? Wait, maybe I misread. Alternatively, the given marks: \( VU = VT \) (one pair of sides), \( \angle VUS=\angle VTS \) (one pair of angles), and \( VS = VS \) (common side). But SAS is when the angle is between the two sides. Wait, actually, the correct theorem here is SAS. Wait, let's check the options. The options are AAS, SSS, SAS, ASA, HL.

Wait, another way: \( VU = VT \) (side), \( \angle UV S=\angle TV S \) (angle), \( VS = VS \) (side). So two sides and the included angle: that's SAS. So the congruence theorem that applies is SAS.

Answer:

SAS (the option corresponding to SAS, depending on the checkbox, but the theorem is SAS)