Which congruence criteria can be
used to prove that the two triangles in the given figure are congruent? Option (A) SSS, option (B) SAS,
option (C) ASA.
In this question, we’re asked for a
congruence criteria. If we look at the options here, we
can see that the S refers to side and the A represents angle. So, let’s look at our two
triangles, 𝐴𝐵𝐶 on the left and triangle 𝐸𝐷𝐹 on the right. We’ll make a note of any
corresponding pairs of angles or sides which are congruent.
In triangle 𝐴𝐵𝐶, we can see that
this angle 𝐴𝐵𝐶 is marked as 104 degrees. The same is true of angle 𝐸𝐷𝐹 in
triangle 𝐸𝐷𝐹. Therefore, we could say that these
two angles are congruent. We can see that angle 𝐴𝐶𝐵 is
22.8 degrees and so is angle 𝐸𝐹𝐷. So, we have another pair of
congruent angles. We can see that there are two sides
which are marked as 7.1, side 𝐴𝐶 on triangle 𝐴𝐵𝐶 and side 𝐸𝐹 on triangle
𝐸𝐷𝐹. Therefore, these sides are
What we’ve shown here is that we
have angle-angle-side, so we could use the angle-angle-side rule to prove
congruence. We could say that triangle 𝐴𝐵𝐶
and triangle 𝐸𝐷𝐹 are congruent using the AAS rule.
A quick reminder that the order of
letters is important when describing congruence. For example, we know that angle 𝐶
in triangle 𝐴𝐵𝐶 is congruent with angle 𝐹 in triangle 𝐸𝐷𝐹. We know that angle 𝐵 and angle 𝐷
are congruent, and angle 𝐴 and 𝐸 are congruent. So, when we look at the answer
options, we see a problem. The AAS rule is not listed as an
option. So, let’s see if we could prove
congruence using another rule too.
We don’t know any additional
information about the length of the sides. So, let’s have a look at the
angles. If we look at angle 𝐵𝐴𝐶 in the
first triangle and angle 𝐷𝐸𝐹 in the second triangle, we could actually work out
the value of these angles by subtracting 104 and 22.8 from 180 degrees, as we know
that there are 180 degrees in total in the triangle. So, both of these angles would be
equal to each other; they’re congruent.
We’ve also just proved that these
two triangles are congruent. Therefore, we know that these third
angles must also be congruent. So therefore, if we take into
account these last three pieces of information, we have two angles and the included
side. Therefore, we have the ASA
rule. So, we’ve shown that these
triangles are congruent using the ASA rule as well, which was the answer given in