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Outside the Box Geometry

# Congruence and Similarity

Two polygons are congruent if all corresponding angles and all corresponding sides between the two figures are equal. In other words, they are the same size and shape.

Two polygons are similar if the ratios of their corresponding side lengths are equal. In other words, similar figures have the same shape but not necessarily the same size.

Which two of the polygons below are definitely congruent?

Triangle X has side lengths $$1, \sqrt{2},$$ and $$2.$$
Triangle Y has side lengths $$\sqrt{2}, 2,$$ and $$2\sqrt{2}.$$

Are triangles X and Y similar?

Siri and Ty have two congruent triangles. Two of the side lengths in Siri's triangle measure 6 and 10. Which set could be two possible side lengths for Ty's triangle?

Dan and Ben both draw right triangles with one side length of 9. Will their triangles definitely be congruent? Note: Right triangles have exactly one $$90^\circ$$ angle.

We can say that a description of a triangle

• defines a unique triangle, if all of the triangles that have all of the specified properties are congruent;

• is underconstrained, if at least two different (non-congruent) triangles can have all of the specified properties;

• is overconstrained, if no triangle exists that has all of the specified properties.

For example, in the previous problem we were given two properties of a triangle: one $$90^\circ$$ angle and one side length of 9.

This description is underconstrained because many non-congruent triangles meet both criteria:

True or False?

Knowing that two of a triangle's angles are $$30^\circ$$ and $$60^\circ$$ uniquely defines a triangle.

Which of the following descriptions uniquely defines a triangle?

Triangles S and T have two sides in common and two angles in common. Therefore, triangles S and T are $$\text{__________}.$$

1. not necessarily either similar or congruent

2. similar but not necessarily congruent

3. both similar and congruent

We have examined many cases regarding what factors can define/constrain a triangle. Let's summarize some of these cases.

Note that not any three parameters will fully define the triangle, even though some sets of three parameters are able to. Here is a summary of the cases that we have examined so far:

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