Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF . Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem.
In order to show that △ABC is congruent to △DEF, we can use a sequence of rigid transformations.
1. Translation: We can translate △ABC to a new position where one of its vertices coincides with the corresponding vertex of △DEF. This ensures that the triangles have the same size and shape.
2. Rotation: By rotating △ABC around its coinciding vertex, we can align another pair of corresponding vertices of the triangles. This rotation preserves the shape and size of the triangles.
3. Reflection: Finally, we can reflect △ABC across a line passing through the coinciding vertex. This step aligns the remaining vertices of the triangles.
Each transformation in this sequence preserves both shape and size, which means that △ABC remains congruent to △DEF throughout the entire process.
This sequence of rigid transformations illustrates the ASA (Angle-Side-Angle) Congruence Theorem. By using a translation, rotation, and reflection in that order, we show that the two triangles have corresponding side lengths in the same order and corresponding angles with the same measures. This confirms that △ABC and △DEF are indeed congruent.
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To show that triangles △ABC and △DEF are congruent, we can use a sequence of rigid transformations. A rigid transformation is a transformation that preserves distance and angles.
1. Translation: We can translate triangle △DEF so that one of its sides coincides with a corresponding side of triangle △ABC. This translation preserves distance and angles, which means that the resulting figure is congruent to the original triangle △DEF.
2. Rotation: Once triangle △DEF is translated, we can rotate it to align another side with a corresponding side of triangle △ABC. We need to determine the angle of rotation that will bring the two triangles into alignment. This rotation further preserves distance and angles, ensuring that the resulting figure is congruent to the previous one.
3. Reflection: Finally, we can reflect the translated and rotated triangle △DEF across a line, which will bring the remaining side into correspondence with the corresponding side of triangle △ABC. Reflection preserves distance and angles, making the final figure congruent to the previous one.
By performing these rigid transformations (translation, rotation, and reflection), we have shown that △ABC is congruent to △DEF. This sequence of transformations demonstrates the ASA Congruence Theorem, which states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the two triangles are congruent.
To show that △ABC is congruent to △DEF, we can use a sequence of rigid transformations. A rigid transformation is a transformation that preserves the shape and size of an object. The sequence of rigid transformations we can use is:
1. Translation: Move triangle △ABC so that one of its sides is coincident with one of the corresponding sides of △DEF. This translation guarantees that the corresponding sides will be congruent because translations preserve lengths.
2. Rotation: Rotate triangle △ABC around the vertex of the coincident side until another side of △ABC matches the corresponding side of △DEF. The angle of rotation will depend on the measure of the corresponding angles. Rotations preserve lengths and angles, so the new position of △ABC will be congruent to △DEF.
3. Reflection: Reflect triangle △ABC across a line passing through one of the sides that has its corresponding side in △DEF. This reflection ensures that the corresponding sides will remain congruent because reflections preserve lengths.
Each of these transformations preserves lengths and angles, which are the properties needed to establish congruence between two triangles.
By using a sequence of Translation, Rotation, and Reflection (in any order), we demonstrate a practical example of the ASA Congruence Theorem. According to the ASA Congruence Theorem, if two angles and their included side of one triangle are congruent to two angles and their included side of another triangle, then the triangles are congruent.
In our example, the translations ensure that two corresponding sides are equal in length, and the rotation and reflection preserve the corresponding angles. Hence, the sequence of transformations shows that △ABC is congruent to △DEF, illustrating the ASA Congruence Theorem.