Determine whether the triangles in the given figure are congruent by applying SSS, SAS, or ASA. If they are congruent, state which of the congruence criteria proves this.
So we’re presented with two triangles, triangles 𝐴𝐵𝐶 and 𝐴 dash 𝐵 dash 𝐶 dash, and asked to determine whether or not they’re congruent. We’re also given three possible congruence criteria. Remember here, S stands for side and 𝐴 stands for angle.
Let’s look at the two diagrams and see what congruence statements we can write down. First, we see that both triangles have a length of three units. This side is 𝐴 dash 𝐵 dash in the first triangle and 𝐴𝐵 in the second, so we have the statement 𝐴 dash 𝐵 dash is equal to 𝐴𝐵. The S in brackets is used to indicate that this is a statement about the length of a side.
Next, we see that both triangles have a length of five units, its side 𝐵 dash 𝐶 dash in the first triangle and side 𝐵𝐶 in the second. So we have the statement 𝐵 dash 𝐶 dash is equal to 𝐵𝐶. And again, the S in brackets indicates that this is a statement about a side.
Finally, we see that both triangles have a length of 3.16 units, side 𝐴 dash 𝐶 dash in the first triangle and side 𝐴𝐶 in the second. So we have the statement 𝐴 dash 𝐶 dash is equal to 𝐴𝐶, and again the S in brackets to indicate it’s a statement about a side.
Now looking at the three statements we’ve made and comparing them to the congruence criteria, we can see that we do have enough information to conclude that these two triangles are congruent. The inclusion of the S’s as we wrote down our congruency statements tells us that it’s the side-side-side, SSS, congruence criteria.
And so we have our answer to the problem: yes, the two triangles are congruent, and this is due to the side-side-side congruence criteria.