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Side-Angle-Side Congruence by basic rigid motions. A geometric realization of a proof in H. Wu’s “Teaching Geometry According to the Common Core Standards”. Given two triangles, ABC and A 0 B 0 C 0 . Assume two pairs of equal corresponding sides with the angle between them equal. B 0. C.

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## Side-Angle-Side Congruence by basic rigid motions

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**Side-Angle-Side Congruence by basic rigid motions**A geometric realization of a proof in H. Wu’s “Teaching Geometry According to the Common Core Standards”**Given two triangles, ABC and A0B0C0.**Assume two pairs of equal corresponding sides with the angle between them equal. B0 C C0 side side angle angle A side B side A0 We want to prove the triangles are congruent.**In other words, given ABC and A0B0C0,**with A = A0, |AB| = |A0B0|, B0 and |AC| = |A0C0|, C C0 side side angle angle A side B side A0 we must give a composition of basic rigid motions that maps ABC exactly onto A0B0C0.**We first move vertex A to A0 by a translation along the**vector from A to A0 B0 C C0 A B A0 translates all points in the plane. Original positions are shown with dashed lines and new positions in red.**Then we use a rotation to bring the horizontal side of the**red triangle (which is the translated image of AB by ) to A0B0. B0 C C0 A B A0**maps the translated image of AB exactly onto A0B0because**|AB| = |A0B0| and translations preserve length. B0 C C0 A B A0**Now we have two of the red triangle’s vertices coinciding**with A0 and B0 of A0B0C0. B0 C C0 A B A0 After a reflection of the red triangle across A0B0, the third vertex will exactly coincide with C0.**Can we be surethis composition of basic rigid motions**(the reflection of the rotation of the translation of the B0 image of ABC) C C0 A B A0 takes C to C0 — and the red triangle with it?**Yes! The two marked angles at A0are equal since basic rigid**motions preserve degrees of angles, and CAB = C0A0B0 is given by hypothesis. B0 C C0 A B A0 A reflection across A0B0 does take C to C0 — and the red triangle with it!**Since basic rigid motions preserve length and since |AC| =**|A0C0|, B0 after a reflection across A0B0, C C0 A B A0 by Lemma 8, the red triangle coincides with A0B0C0. The triangles are congruent. Our proof is complete.**Given two triangles with two pairs of equal sides and an**included equal angle, a composition of basic rigid motions B0 B0 (translation, rotation, and reflection) C C0 C0 A B A0 A0 maps the image of one triangle onto the other. Therefore, the triangles are congruent.

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