Which is the correct congruence from this figure? Option (A) triangle 𝐴𝐵𝐷 is congruent to triangle 𝐴𝐷𝐶. Option (B) triangle 𝐵𝐷𝐴 is congruent to triangle 𝐶𝐷𝐴. Option (C) triangle 𝐴𝐵𝐷 is congruent to triangle 𝐶𝐷𝐴. Option (D) triangle 𝐴𝐷𝐵 is congruent to triangle 𝐶𝐷𝐴.
Here we can see that there are two triangles, 𝐴𝐵𝐷 and 𝐴𝐶𝐷. As we’re told that there is some sort of congruence between these two triangles, it would be sensible to list any angles or pairs of sides which are congruent. So let’s take our two triangles, 𝐴𝐵𝐷 and 𝐴𝐶𝐷. The angle at 𝐴𝐵𝐷 is marked as congruent with the angle at 𝐴𝐶𝐷. So that’s a pair of congruent angles. In triangle 𝐴𝐵𝐷, we have this side 𝐵𝐷 marked as the same length as side 𝐶𝐷 in triangle 𝐴𝐶𝐷. We can tell this from the one marking on each of these lines. So that’s a pair of congruent sides. We have another pair of congruent angles. Angle 𝐵𝐷𝐴 is congruent with angle 𝐶𝐷𝐴. So that’s another pair of congruent angles.
This means that we have found two pairs of corresponding angles congruent and an included pair of corresponding sides congruent. We could then say that these triangles are congruent using the ASA or angle-side-angle rule. We might also notice that we have a line 𝐴𝐷 which is common to both sides, meaning that we’ve actually found another pair of congruent sides. So we could have alternatively said that these triangles are congruent using the side-angle-side rule.
The question has asked us to select the correct congruence for the figure. So what does that mean? You may recall that the order of the letters is important when we’re writing congruence. If we look at the side 𝐴𝐵 in this top triangle, that’s congruent with our side 𝐴𝐶 on the lower triangle. The length going from 𝐵𝐷 is congruent with the length 𝐶𝐷. And so we could say that triangle 𝐴𝐵𝐷 is congruent with triangle 𝐴𝐶𝐷.
We can also write the letters in a different way. For example, we could start with 𝐵𝐷 on triangle 𝐴𝐵𝐷, but then we’d also have to start with the letters 𝐶𝐷 in triangle 𝐴𝐶𝐷. So we could write the relationship that triangle 𝐵𝐷𝐴 is congruent to triangle 𝐶𝐷𝐴.
We could also start with our letter 𝐷 and say that triangle 𝐷𝐴𝐵 is congruent to triangle 𝐷𝐴𝐶. Any of these would be a correct congruence for the figure. However, the only one listed in the answer options is that given in option (B) triangle 𝐵𝐷𝐴 is congruent to triangle 𝐶𝐷𝐴.