# SAS similarity

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One of the main $3$ main theorems for proving the similarity between $2$ triangles. Similarity (specifically for triangles here) means all the angles in these triangles are equal and all sides are proportional to each other. SAS similarity is a similarity theorem stating that if there are two proportional sides and an angle in between it is equal, then they are similar by SAS similarity.

Example Problem: There are two isosceles triangles, $\triangle ABC$ and $\triangle DEF$. $m\angle B = m\angle E= 32^\circ$, and $\overline{AB} = 9$, $\overline{DE} = 3$, $\overline{BC} = 2$, $\overline{EF} = 6$. What are the measures of $m\angle BAC, m\angle BCA, m\angle EDF$ and $m\angle EFD$ (All of these are base angles for their respective triangles)?

Example Solution: Since sides $\overline{AB}$ and $\overline{DE}$ with their common ratio being $3$, and $\overline{EF}$ and $\overline{BC}$ also having a common ratio of $3$, and both of them sharing a common angle $\angle B$ and $\angle E$, $\triangle ABC$ and $\triangle DEF$ are similar by SAS Similarity.

Now, if $2$ triangles are similar, all their angles are equal, which means that $m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD$ since they are all base angles of similar triangles. If one of the non-base angles is $32^\circ$, it means that $\frac{(180-32)}{2}$, which is equal to $\frac{148}{2}$, which is $74^\circ$. So $m\angle BAC = m\angle BCA = m\angle EDF = m\angle EFD = 74^\circ$.