Video: Using Triangle Congruence Criteria to Establish Congruence

Which congruence criteria can be used to prove that the two triangles in the figure are congruent? [A] SSS [B] SAS [C] ASA

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Video Transcript

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, weโ€™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, letโ€™s look at our two triangles, ๐ด๐ต๐ถ on the left and triangle ๐ธ๐ท๐น on the right. Weโ€™ll make a note of any corresponding pairs of angles or sides which are congruent.

In triangle ๐ด๐ต๐ถ, we can see that this angle ๐ด๐ต๐ถ is marked as 104 degrees. The same is true of angle ๐ธ๐ท๐น in triangle ๐ธ๐ท๐น. Therefore, we could say that these two angles are congruent. We can see that angle ๐ด๐ถ๐ต is 22.8 degrees and so is angle ๐ธ๐น๐ท. So, we have another pair of congruent angles. We can see that there are two sides which are marked as 7.1, side ๐ด๐ถ on triangle ๐ด๐ต๐ถ and side ๐ธ๐น on triangle ๐ธ๐ท๐น. Therefore, these sides are congruent.

What weโ€™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 ๐ด๐ต๐ถ and triangle ๐ธ๐ท๐น 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 ๐ถ in triangle ๐ด๐ต๐ถ is congruent with angle ๐น in triangle ๐ธ๐ท๐น. We know that angle ๐ต and angle ๐ท are congruent, and angle ๐ด and ๐ธ are congruent. So, when we look at the answer options, we see a problem. The AAS rule is not listed as an option. So, letโ€™s see if we could prove congruence using another rule too.

We donโ€™t know any additional information about the length of the sides. So, letโ€™s have a look at the angles. If we look at angle ๐ต๐ด๐ถ in the first triangle and angle ๐ท๐ธ๐น 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; theyโ€™re congruent.

Weโ€™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, weโ€™ve shown that these triangles are congruent using the ASA rule as well, which was the answer given in option (C).

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