![]() ![]() These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent. Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of “same shape” and “scale factor” developed in the middle grades. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. The four most common reflections are performed over the following lines of reflection: the $x$-axis, the $y$-axis, $y =x$, and $y =-x$.In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. However, the orientation of the points or vertices changes when reflecting an object over a line of reflection. In fact, in reflection, the angle measures of the objects, parallelism, and side lengths will remain intact. The distances between the vertices of the triangles from the line of reflection will always be the same. The graph above showcases how a pre-image, $\Delta ABC$, is reflected over the horizontal line of reflection $y = 4$. This makes reflection a rigid transformation. ![]() ![]() When learning about point and triangle reflection, it has been established that when reflecting a pre-image, the resulting image changes position but retains its shape and size. In reflection, the position of the points or object changes with reference to the line of reflection. Once we’ve established their foundations, it will be easier to work on more complex examples of rigid transformations. We’ll explore different examples of reflection, translation and rotation as rigid transformations. It’s time to explore these three examples of basic rigid transformations first. This makes this transformation a rigid transformation.
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