Two sides and the angle between
Two ratios can agree, and the triangles can still be different shapes.
- Two sides of a triangle: 4 and 6, with 50° between.
- A second triangle: sides 6 and 9, also 50° between.
- 6/4 and 9/6 are both 3/2.
- Ratio matches, angle matches: similar.
The side-angle-side test
Suppose two pairs of sides are in one ratio. The angles between those pairs must be equal. Then the triangles are similar. This is the side-angle-side test, or SAS.
Two pairs of sides are in one ratio. The angles between those sides are equal. Then the triangles are similar.
The word between matters. It is the angle at the corner where the two sides meet. Pair the sides in order. Triangle one's 4 matches triangle two's 6. Triangle one's 6 matches triangle two's 9.
Why the angle matters
Take sides 6 and 9 in one triangle. Take sides 8 and 12 in another. Both ratios are 4/3.
Make the angle between 6 and 9 wide. Make the angle between 8 and 12 narrow. Then the third sides are not in the ratio 4/3. So the triangles are not similar.
Slide the angle between the sides 6 and 9.
Loading lab…