Transformations—The Next Level
a. Find the measure of each lettered angle in the figure below.
b. Given: 𝑚∠𝐶𝐷𝐸 = 𝑚∠𝐵𝐴𝐶
Prove: 𝑚∠𝐷𝐸𝐶 = 𝑚∠𝐴𝐵𝐶
Explaining how to transform figures without the benefit of a coordinate plane can be difficult without some important vocabulary. Let’s review.
The word transformation has a specific meaning in geometry. A transformation 𝐹 of the plane is a function that assigns to each point 𝑃 of the plane a unique point 𝐹(𝑃) in the plane. Transformations that preserve lengths of segments and measures of angles are called _____. A dilation is an example of a transformation that preserves _____ measures but not the lengths of segments. In this lesson, we work only with rigid transformations. We call a figure that is about to undergo a transformation the _____, while the figure that has undergone the transformation is called the _____.
Using the figures above, identify specific information needed to perform the rigid motion shown.
For a rotation, we need to know: _____
For a reflection, we need to know: _____
For a translation, we need to know: _____
We have now done some work with all three basic types of rigid motions (rotations, reflections, and translations). At
this point, we need to state our assumptions as to the properties of basic rigid motions:
a. Any basic rigid motion preserves lines, rays, and segments. That is, for a basic rigid motion of the plane, the image of a line is a line, the image of a ray is a ray, and the image of a segment is a segment.
b. Any basic rigid motion preserves lengths of segments and measures of angles. Relevant Vocabulary
BASIC RIGID MOTION: A basic rigid motion is a rotation, reflection, or translation of the plane. Basic rigid motions are examples of transformations. Given a transformation, the image of a point 𝐴 is the point the transformation maps 𝐴 to in the plane.
DISTANCE-PRESERVING: A transformation is said to be distance-preserving if the distance between the images of two points is always equal to the distance between the pre-images of the two points.
ANGLE-PRESERVING: A transformation is said to be angle-preserving if (1) the image of any angle is again an angle and (2) for any given angle, the angle measure of the image of that angle is equal to the angle measure of the pre-image of that angle.
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