Video Transcript
Given that 𝐸 is the midpoint of line segment 𝐴𝐶 in the given figure, without
referencing angles, which congruence criterion could you use to prove triangles
𝐴𝐵𝐸 and 𝐶𝐵𝐸 are congruent?
In the figure, we can observe that there is a larger triangle, 𝐴𝐵𝐶, which is split
into two smaller triangles, 𝐴𝐵𝐸 on the left and triangle 𝐶𝐵𝐸 on the right. We are given the information that 𝐸 is the midpoint of the line segment 𝐴𝐶. And that means that the two line segments 𝐴𝐸 and 𝐸𝐶 must be equal in length or
congruent. We need to determine how we might prove that triangle 𝐴𝐵𝐸 is congruent to triangle
𝐶𝐵𝐸. And we must do that without using any angle properties. So let’s continue to see if there are any other lengths which might be congruent.
From the markings on the diagram, we can see that the line segments 𝐴𝐵 and 𝐶𝐵 are
marked as congruent, which would also indicate that the larger triangle 𝐴𝐵𝐶 is an
isosceles triangle. Now, one way in which we can prove that two triangles are congruent is if there are
three pairs of corresponding sides congruent. We have shown there are two pairs. So can we show the third pair? And the answer is yes, since line segment 𝐵𝐸 appears in both triangles. If we were writing a formal proof, we could either write that 𝐵𝐸 is a common side
in the triangles or that 𝐵𝐸 equals 𝐵𝐸. If two triangles share a common side, then we know that it will be congruent in both
triangles.
Therefore, there are three pairs of congruent sides. And so we can prove triangles 𝐴𝐵𝐸 and 𝐶𝐵𝐸 are congruent using the SSS, or
side-side-side, congruency criterion.
