The SAS (Side-Angle-Side) similarity criterion is a method used to establish the similarity of two triangles based on the lengths of their sides and the measure of the included angle. Here’s how each point you've provided relates to triangle similarity:
a. All three sides lengths are given and proportional
This is actually known as the SSS (Side-Side-Side) similarity criterion, rather than SAS. If all three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. In other words, if the ratios of the corresponding sides of the two triangles are equal (resulting in the same scale factor), then the triangles are similar.
b. Two corresponding side lengths are proportional and the included angle is equal
This describes the SAS similarity criterion. If two sides of one triangle are in proportion to two sides of another triangle, and the angle between those two sides is congruent, then the triangles are similar. The corresponding angles will also be equal due to the properties of similar triangles.
c. Given a scale factor
If you know the scale factor between two triangles, you can determine that the triangles are similar — provided the scale factor is consistent for all corresponding sides. However, simply knowing the scale factor is not enough unless you also prove that the angles are congruent or you have a confirmation that the triangles satisfy either the SSS or SAS criteria.
d. At least two angles of the triangles are the same
This is known as the AA (Angle-Angle) similarity criterion. If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. This is because the third angle must also be congruent (the sum of angles in a triangle is always 180 degrees), thus confirming that all corresponding angles are equal.
Summary
- SAS Criterion: Two triangles are similar if two sides of one triangle are proportional to two sides of another triangle and the included angle between those two sides is equal.
- Other Criteria: SSS for side ratios, AA for angle congruence.
- Scale factors confirm similarity but must relate to consistent proportionality of sides and angle measures.
To use the SAS criterion effectively, confirm both that the sides are proportional and that the included angles are congruent, thereby establishing similarity between the two triangles.