Determine whether the triangles in the given figure are congruent, and if they are, state which of the congruence criteria proves this.
So we’ve been given a diagram of two triangles, triangle 𝐴𝐵𝐶 and triangle 𝐴 dash 𝐵 dash 𝐶 dash. We’ve been given various pieces of information about these two triangles. And at a glance, we can see that the values are the same. However, we need to look at the congruence criteria to formally prove whether these two triangles are or aren’t congruent.
Let’s begin by looking at the side of length 2.53 that appears in both triangles. This is side 𝐴𝐵 in the first triangle and side 𝐴 dash 𝐶 dash in the second triangle. So we have that 𝐴𝐵 is congruent to 𝐴 dash 𝐶 dash, and the S in brackets indicates that this is a side.
Next, let’s look at the angle of 60.34 degrees that both triangles have. This is angle 𝐵 in the first triangle and angle 𝐶 dash in the second triangle. So we have that angle 𝐵 is congruent to angle 𝐶 dash, and the A in brackets indicates that this is an angle.
Next, let’s look at the side that measures 3.68 units. This is side 𝐵𝐶 in the first triangle and side 𝐶 dash 𝐵 dash in the second triangle. So we have that 𝐵𝐶 is congruent to 𝐶 dash 𝐵 dash. Again, the S in brackets indicates that we’re referring to a side.
So let’s look at what we’ve shown. We’ve shown that these two triangles have two sides and an angle in common. But more specifically, it’s the included angle, the angle that’s between the two known sides. This is sufficient to prove that the two triangles are congruent to each other. By looking at the letters in brackets, we can see which congruence criteria we’ve used, side angle side is the SAS congruency criteria.
So our answer to the problem is that yes, these two triangles are congruent, and the congruency criteria used to prove this is the side angle side criteria.