Integrated Math 2 / Unit 6: Similarity
6.3 Triangle Similarity: AA, SSS, SAS
Notes
You do not need all six measurements to show triangles are similar.
- AA: two pairs of congruent angles.
- SSS: all three pairs of sides proportional.
- SAS: two pairs of sides proportional and the included angles congruent.
- Side-side-angle (SSA) is NOT a valid test.
- Example: sides $4, 6, 8$ and $6, 9, 12$ share the ratio $1.5$, so they are similar by SSS.
Practice
Triangle 1 has angles $50°$ and $60°$. Triangle 2 has angles $60°$ and $70°$. Are they similar?
The third angles are $70°$ and $50°$, so both triangles have angles $50°, 60°, 70°$.
Triangles with sides $4, 6, 8$ and $6, 9, 12$ are similar by which criterion?
$\displaystyle \frac{6}{4} = \frac{9}{6} = \frac{12}{8} = 1.5$, so all three sides are proportional.
One triangle has sides 5 and 8 with an included angle of $40°$. Another has sides 10 and 16 with an included angle of $40°$. Which is true?
$\displaystyle \frac{10}{5} = \frac{16}{8} = 2$ and the included angles match, so SAS similarity applies.
Ready for more? Take the Unit 6 test.