Determine whether the triangles in the given figure are congruent, and if they are, state which of the congruence criteria proves this.
In this question, we need to establish if triangle 𝐴𝐵𝐶 is congruent with triangle 𝐴 prime 𝐵 prime 𝐶 prime. We can recall that congruent means the same shape and size. And when we’re dealing with congruence in triangles, there are a few congruence criteria that we can use. If we can demonstrate, for instance, that there are three pairs of corresponding sides congruent, then the triangle would be congruent. The angle-side-angle rule means that two pairs of corresponding angles and the included side are congruent. Angle-angle-side means two pairs of angles and any side. The side-angle-side rule would show that two pairs of corresponding sides and the included angle are congruent.
The final criteria, the RHS rule means that we need to show that there’s a right angle in two triangles. The hypotenuse on both triangles is congruent, and there’s another pair of corresponding sides congruent. So let’s have a look at our two triangles and see if we can find any congruent angles or sides. Looking at triangle 𝐴𝐵𝐶, we can see that at angle 𝐴, it’s marked at 51.54 degrees. And there’s a congruent angle in triangle 𝐴 prime 𝐵 prime 𝐶 prime. It’s the angle 𝐴 prime. The line 𝐴𝐶 is marked as 4.27 units, and there’s a congruent line on triangle 𝐴 prime 𝐵 prime 𝐶 prime. It’s the line 𝐴 prime 𝐶 prime. As both of these lines are the same length, we can then say that they’re equal.
Finally, we can see that the angle at 𝐶 is equal to the angle at 𝐶 prime on triangle 𝐴 prime 𝐵 prime 𝐶 prime. If you look at what we’ve noted, we’ve found two pairs of corresponding congruent angles and we’ve also found a pair of corresponding sides congruent. The side that we found is between the two angles. Therefore, we can use the ASA rule to show congruence. We can therefore answer the question that yes, these two triangles are congruent, and we used the ASA or angle-side-angle congruence criteria.