in similarity postulates as they were in congruence theorems. Each[U+0080][U+009C]A[U+0080][U+009D]in a
similarity shortcut stands for one pair of congruent corresponding angles in the triangles.
TheS′s represent a different requirement in similarity postulates then they did in congruence postulates and theorems.
Congruent triangles have congruent sides, but similar triangles have proportional sides. Each[U+0080][U+009C]S[U+0080][U+009D]
is a similarity postulate represents a ratio of corresponding sides. Once the ratios (two for SAS and three for SSS)
are written, equality of the ratios must be verified. If the ratios are equal, the sides in question are proportional, and
the postulate can be applied.
It is sometimes hard for student to adjust to this new side requirement. They have done so much work with congruent
triangles that it is easy for them to slip back into congruent mode. Warn them not to fall into the old way of thinking.
TABLE2.12:
TriangleCongruence Postulates and Theorems TriangleSimilarity Postulates
[U+0080][U+009C]S[U+0080][U+009D] ↔congru-
ent sides[U+0080][U+009C]S[U+0080][U+009D] ↔propor-
tional sides
SSS AA
SAS SSS
ASA SAS
AAS
HLOnly Three Similarity Postulates –Students will sometimes try to use ASA, or other congruence theorems to show
that two triangles are similar. Bring it to their attention that there are only three postulates for similarity, and that
they do not all have the same side and angle combinations as congruence postulates or theorems.
Proportionality Relationships
Similar Triangles Formed by an Interior Parallel Segment –Students frequently are presented with a triangle
that contains a segment that is parallel to one side of the triangle and intersects the other two sides. This segment
creates a smaller triangle in the tip of the original triangle. There are two ways to consider this situation. The two
triangles can be considered separately, or the Triangle Proportionality Theorem can be applied.
(1) Consider the two triangles separately.
The original triangle and the smaller triangle created by the parallel segment are similar as seen in the proof of the
Triangle Proportionality theorem. One way students can tackle this situation is to draw the triangles separately and
use proportions to solve for missing sides. The strength of this method is that it can be used for all three sides of the
triangles. Students need to be careful when labeling the sides of the larger triangle; often the lengths will be labeled
as two separate segments and the students will have to add to get the total length.
(2) Use the Triangle Proportionality theorem.
When using this theorem it is much easier to setup the proportions, but there is the limitation that the theorem can
not be used to find the lengths of the parallel segments.
Ideally student will be able to identify the situations where each method is the most efficient, and apply it. This may
not happen until the students have had some experience with these types of problem. It is best to have students use
method (1) at first, then after they have worked a few exercises on their own, they can use (2) as a shortcut in the
appropriate situations.
Additional Exercises:
- 4 ABChas pointEonAB, andFonBCsuch thatEFis parallel toAC.
2.7. Similarity