Geometry!!!

Quincy suspects that the two triangles above are similar, but there is not enough information to prove this. This is because \(\triangle JKL\) only has two defined side lengths, which is not enough to determine the other unknown angles or sides. 

 

2a) The SAS Similarity Theorem proves the similarity of two triangles by showing that the following is true:

 

• Both side lengths of one triangle are proportional to the side lengths of the other triangle.
• The included angle in one triangle is congruent to the corresponding angle of the other triangle.

Quincy would need the measure of \(\angle K\). This is the included angle of \(\overline{JK}\text{ and }\overline{KL}\) . We can determine the measure of the corresponding angle, \(\angle Q\) , by using the Law of Cosines. 

 

2b) The SSS Similarity Theorem proves the similarity of two triangles by showing that the following is true:

 

• All three side lengths of one triangle are proportional to the corresponding sides of another triangle.

Quincy would need the length of \(\overline{JL}\) . We already know the lengths of the other two side lengths of \(\triangle JKL\) and all the side lengths of \(\triangle PQR\), so this is the only extra bit of information needed to prove similarity. 

 

2c) The AA Postulate proves the similarity of two triangles by showing that the following is true:

 

• Two corresponding angles of the two triangles are congruent. 

Of course, Quincy would need to know the measure of two angles before proving their congruency, so it is an unlikely candidate to prove their similarity. Yes, it is possible to do, but it would be quite unnecessary.