Let the third triangle be, an image of under an isometry. We show that if a third triangle exists, and is congruent to it, then is also congruent to it. In proving the theorem, we will use the transitive property of congruence. If in two triangles, three sides of one are congruent to three sides of the other, then the two triangles are congruent. Now that we finished the prerequisite, we now prove the theorem. It says that for any real numbers, , and, if and, then. Let us recall the transitive property of equality of real numbers. Thus, we say that a kite is reflection-symmetric. Mirroring an image or reflection preserves distance. The diagonal is a line of symmetry of the kite. Properties of KitesĪ kite is a polygon with two distinct pairs of congruent sides. Also, each object in the image has exactly one preimage. Each object in the preimage has exactly one image. Notice that there is a 1-1 mapping between the objects in the preimage and the objects in the image. In the isometry above, the preimage is mapped onto the image. Sliding or translation is a form of isometry, a type of mapping that preserves distance. Clearly, when you side a figure, the size and shape are preserved, so clearly, the two triangles are congruent. In the figure below, is slid to the right forming. If you are familiar with these concepts, you can skip them and go directly to the proof. These concepts are isometries particulary reflection and translation, properties of kites, and the transitive property of congruence. Recall that the theorem states that if three corresponding sides of a triangle are congruent, then the two triangles are congruent.īefore proving the SSS Congruence theorem, we need to understand several concepts that are pre-requisite to its proof. In this post, we are going to prove the SSS Congruence Theorem.
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