8.4 SSS and SAS Similarity
Notes
Sides can also prove similarity.
- SSS Similarity: all three pairs of corresponding sides are proportional.
- SAS Similarity: two pairs of sides are proportional and the included angles are congruent.
- Check by writing each ratio, shortest to shortest, middle to middle, longest to longest.
- Example: sides 4, 6, 8 and 6, 9, 12 have ratios $\displaystyle \frac{6}{4} = \frac{9}{6} = \frac{12}{8} = 1.5$, so the triangles are similar by SSS.
Practice
Which triangle is similar to a triangle with sides 5, 7 and 9?
Every side is multiplied by 3.
$\triangle PQR$ has $PQ = 4$, $PR = 6$ and $m\angle P = 50^\circ$. $\triangle STU$ has $ST = 10$, $SU = 15$ and $m\angle S = 50^\circ$. Which is true?
$\displaystyle \frac{10}{4} = \frac{15}{6} = 2.5$ and the included angles are congruent.
Triangles with sides 3, 4, 6 and 6, 8, 10. Are they similar?
$\displaystyle \frac{6}{3} = \frac{8}{4} = 2$, but $\displaystyle \frac{10}{6} \ne 2$. All three ratios must match.
Ready for more? Take the Unit 8 test.