Question

fill in the blanks to prove ( \triangle abccong\triangle def ).

• i can translate ( \triangle abc ) so that point ( a ) goes to point.

• if i rotate this image around point ( d ), i know that point ( b ) will align with point ( e ) because ( congoverline{de} ).

• i can reflect this image over ( overline{de} ).

• i know that ( angle a ) will align with and ( angle b ) will align with because angle measure is preserved with rigid transformations.

• after the reflection, i know that point ( c ) will stack onto point.

• after proving all of the corresponding vertices align, i know that ( \triangle abccong\triangle def ).

Explanation:

Step1: First blank

Since we want to prove \(\triangle ABC\cong\triangle DEF\), and we are using rigid - transformations (translation, rotation, reflection). For the translation, we want to map \(A\) to \(D\) (the corresponding vertex in \(\triangle DEF\)).

Step2: Second blank

We know that in congruent triangles, corresponding sides are equal. In \(\triangle ABC\) and \(\triangle DEF\), \(AB\) corresponds to \(DE\). So, \(AB\cong DE\) (by the definition of congruent triangles, if \(\triangle ABC\cong\triangle DEF\), then \(AB = DE\) in length, and for the rotation to make \(B\) align with \(E\), we use the fact that \(AB\cong DE\)).

Step3: Third and fourth blanks

In congruent triangles, corresponding angles are equal. \(\angle A\) corresponds to \(\angle D\) and \(\angle B\) corresponds to \(\angle E\) (by the definition of congruent triangles \(\triangle ABC\cong\triangle DEF\), so \(\angle A=\angle D\) and \(\angle B = \angle E\) in measure, and rigid - transformations (translation, rotation, reflection) preserve angle measures).

Step4: Fifth blank

After mapping \(A\) to \(D\), \(B\) to \(E\) using translation and rotation, and then reflecting over \(DE\), since \(\triangle ABC\cong\triangle DEF\), the remaining vertex \(C\) will map to \(F\) (by the definition of congruent triangles, all corresponding vertices must match).

Answer:

1. \(D\)
2. \(AB\)
3. \(\angle D\)
4. \(\angle E\)
5. \(F\)