Question

guided practice
determine if the two figures are congruent by using transformations.
explain your reasoning. (examples 1 and 2)
1.

Explanation:

Step1: Recall congruence by transformation

Two figures are congruent if one can be mapped onto the other by rigid - motions (translations, rotations, reflections).

Step2: Analyze possible transformation

We can try to find a sequence of translations, rotations, and reflections that maps $\triangle LMN$ onto $\triangle VIX$. First, we can check if a translation can align one of the vertices. Let's assume we can translate $\triangle LMN$ so that point $L$ coincides with point $V$. Then we can check if a rotation around the common - vertex (after translation) and/or a reflection can make the two triangles coincide. If we can find such a sequence of rigid - motions, the two triangles are congruent. If not, they are not congruent.
Let's assume we translate $\triangle LMN$ so that $L$ moves to $V$. Then we observe the orientation of the triangles. We can see that we can rotate $\triangle LMN$ (after translation) around the new position of $L$ (which is now at $V$) to align the sides and then, if necessary, perform a reflection. In this case, we can translate $\triangle LMN$ horizontally and vertically so that $L$ coincides with $V$. Then we can rotate $\triangle LMN$ counter - clockwise about point $V$ to align the sides and angles. Since we can map $\triangle LMN$ onto $\triangle VIX$ using a translation and a rotation (both rigid - motions), the two triangles are congruent.

Answer:

The two triangles $\triangle LMN$ and $\triangle VIX$ are congruent because we can map $\triangle LMN$ onto $\triangle VIX$ using a translation and a rotation, which are rigid - motions. Rigid - motions preserve the shape and size of a figure, so the two figures are congruent.