Two triangles are labeled upper A upper B upper C, and upper F upper E upper D.
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.
To show that △ABC ≅ △DEF, we can use a sequence of rigid transformations: translation, rotation, and another translation.
1. Translation: Move triangle △ABC so that point A coincides with point D.
- This translation preserves both shape and size, as all points of triangle △ABC are shifted the same distance and in the same direction.
- Now, we have △ABC and △DEF with A and D coinciding.
2. Rotation: Rotate triangle △ABC counter-clockwise so that side AB coincides with side DE.
- This rotation preserves both shape and size, as all points of triangle △ABC are rotated an equal angle and around the same center of rotation.
- Now, we have △ABC and △DEF with sides AB and DE aligned.
3. Translation: Move triangle △ABC so that point B coincides with point F.
- This translation preserves both shape and size, as all points of triangle △ABC are shifted the same distance and in the same direction.
- Now, we have △ABC and △DEF with sides AB and DE aligned and points B and F coinciding.
By applying these sequence of rigid transformations, we have shown that △ABC ≅ △DEF.
This sequence of rigid transformations illustrates the ASA (Angle-Side-Angle) Congruence Theorem. The theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
In our case, we have:
- ∠BAC ≅ ∠EDF (by the rotation step, which preserves angle measure),
- AB ≅ DE (by the rotation step, which preserves length),
- ∠CAB ≅ ∠FDE (by the rotation step, which preserves angle measure).
Therefore, by the ASA Congruence Theorem, we conclude that △ABC ≅ △DEF.