When triangles are congruent (identical), one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and corresponding angles will be congruent.
The good news is that when trying to verify that two triangles congruent , it is not necessary to show that all six of these facts to be true. There are certain ordered combinations of these facts that are sufficient to verify triangles to be congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.
The following ordered combinations of the congruent triangle facts will NOT be sufficient to prove triangles congruent. Let's see why these combinations DO NOT work!
Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length. The side lengths of two similar triangles are proportional. That is, if Δ U V W is similar to Δ X Y Z , then the following equation holds: U V X Y = U W X Z = V W Y Z This common ratio is called the scale factor . The symbol ∼ is used to indicate similarity.
Example: Δ U V W ∼ Δ X Y Z . If U V = 3 , V W = 4 , U W = 5 and X Y = 12 , find X Z and Y Z . Draw a figure to help yourself visualize.
Write out the proportion. Make sure you have the corresponding sides right. 3 12 = 5 X Z = 4 Y Z The scale factor here is 3 12 = 1 4 . Solving these equations gives X Z = 20 and Y Z = 16 . The concepts of similarity and scale factor can be extended to other figures besides triangles. |