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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?

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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?

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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.

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Which of the following descriptions **uniquely defines** a triangle?

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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

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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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