So far, we have mainly looked at congruence for polygons. Polygons are special because they are determined by line segments. These line segments give polygons easily defined distances and angles to measure and compare. For a more complex shape with curved sides, the situation is a little different (unless the shape has special properties such as being a circle). The focus here is on the fact that the distance between any pair of corresponding points of congruent figures must be the same. Because there are too many pairs of points to consider, this is mainly a criterion for showing that two figures are not congruent: that is, if there is a pair of points on one figure that are a different distance apart than the corresponding points on another figure, then those figures are not congruent.
One of the mathematical practices that takes center stage in this lesson is MP6. For congruent figures built out of several different parts (for example, a collection of circles) the distances between all pairs of points must be the same. It is not enough that the constituent parts (circles for example) be congruent: they must also be in the same configuration, the same distance apart. This follows from the definition of congruence: rigid motions do not change distances between points, so if figure 1 is congruent to figure 2 then the distance between any pair of points in figure 1 is equal to the distance between the corresponding pair of points in figure 2.
- Determine whether shapes are congruent by measuring corresponding points.
- Draw and label corresponding points on congruent figures.
- Justify (orally and in writing) that congruent figures have equal corresponding distances between pairs of points.
Let’s find ways to test congruence of interesting figures.
- I can use distances between points to decide if two figures are congruent.
One figure is congruent to another if it can be moved with translations, rotations, and reflections to fit exactly over the other.
In the figure, Triangle A is congruent to Triangles B, C, and D. A translation takes Triangle A to Triangle B, a rotation takes Triangle B to Triangle C, and a reflection takes Triangle C to Triangle D.
A right angle is half of a straight angle. It measures 90 degrees.
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