A student claims that two triangles are congruent Select EACH process that the student can take to justify the claim a Show that the corresponding sides of the two triangles are congruent B Show that the corresponding sides of the two triangles are proportional. C Show that there exists a sequence of rigid motions that carries one triangle onto the other. D. Show that two corresponding angles and the included side of the two triangles are congruent. E. Show that there is a sequence of rigid motions and dilations that carries one triangle onto the other

## Step 1<br />The problem is about the criteria for two triangles to be congruent. Congruent triangles are triangles that have the same size and shape. This means that their corresponding sides are equal and their corresponding angles are equal.<br /><br />## Step 2<br />Option A suggests showing that the corresponding sides of the two triangles are congruent. This is a valid method to prove that two triangles are congruent. If all the corresponding sides of two triangles are equal, then the triangles are congruent.<br /><br />## Step 3<br />Option B suggests showing that the corresponding sides of the two triangles are proportional. This is not a valid method to prove that two triangles are congruent. If the sides of two triangles are proportional, it means the triangles are similar, not congruent. Similar triangles have the same shape but not necessarily the same size.<br /><br />## Step 4<br />Option C suggests showing that there exists a sequence of rigid motions that carries one triangle onto the other. This is a valid method to prove that two triangles are congruent. Rigid motions (translations, rotations, and reflections) preserve the size and shape of figures. If one triangle can be carried onto the other by a sequence of rigid motions, it means the two triangles are congruent.<br /><br />## Step 5<br />Option D suggests showing that two corresponding angles and the included side of the two triangles are congruent. This is a valid method to prove that two triangles are congruent. This is known as the Angle-Side-Angle (ASA) congruence criterion. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.<br /><br />## Step 6<br />Option E suggests showing that there is a sequence of rigid motions and dilations that carries one triangle onto the other. This is not a valid method to prove that two triangles are congruent. Dilations change the size of a figure, so even if a sequence of rigid motions and dilations can carry one triangle onto the other, the triangles are not congruent but similar.