Question

which of the following congruence theorems can be used to prove that the two triangles are congruent?

Explanation:

Step1: Analyze the diagram

From the diagram, we can see that $\angle GJH$ and $\angle KJH$ are right angles (since they form a linear pair and a semicircle, so each is $90^\circ$). Also, $HJ$ is a common side to both triangles $\triangle GJH$ and $\triangle KJH$. We can assume that $\angle G=\angle K$ (from the markings on the angles at $G$ and $K$).

Step2: Identify the congruence theorem

We have two angles and a non - included side? Wait, no. Wait, in right - angled triangles, if we have a right angle, a hypotenuse, and a leg, it's HL, but here, let's re - examine. Wait, the triangles are $\triangle GHJ$ and $\triangle KHJ$. $\angle G=\angle K$, $\angle GJH=\angle KJH = 90^\circ$, and $HJ$ is common. So by the AAS (Angle - Angle - Side) congruence theorem, or if we consider the right angle, angle, and side, AAS or maybe AAS. Wait, AAS states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, the triangles are congruent. Here, $\angle G=\angle K$, $\angle GJH=\angle KJH$, and $HJ = HJ$ (common side, non - included between the two angles). So the AAS (Angle - Angle - Side) congruence theorem can be used. Also, since it's a right - angled triangle, we can also think of AAS as a valid theorem here. Alternatively, if we consider that $\angle G=\angle K$, $\angle GJH=\angle KJH = 90^\circ$, and $HJ$ is common, AAS applies.

Answer:

The congruence theorem that can be used is AAS (Angle - Angle - Side) (or if considering right - angled triangles, AAS is applicable here. If we assume the triangles are right - angled, AAS or also, since $\angle G=\angle K$, $\angle GJH=\angle KJH$, and $HJ$ is common, AAS is the theorem).