Question

in the figure, \\( \triangle abc \\) and \\( \triangle edc \\) are shown. given: \\( \angle a \cong \angle e \\) and \\( \angle bca \cong \angle dce \\).
three transformations, when performed in series on \\( \triangle abc \\), will show that the two triangles are similar. move statements to the lines to show the steps.
step 1: clockwise rotation of \\( 90^{\circ} \\) about point \\( c \\)
step 2:
step 3:
reflection across \\( \overline{ac} \\) reflection across \\( \overline{bc} \\)

Explanation:

Step 1: Clockwise rotation of \(90^{\circ}\) about point \(C\)

This rotation will re - orient \(\triangle ABC\) so that the angles \(\angle BCA\) and \(\angle DCE\) start to align in a more comparable way.

Step 2: Reflection across \(\overline{AC}\)

After rotation, a reflection across \(\overline{AC}\) will further align the triangles. Since \(\angle A\cong\angle E\) and \(\angle BCA\cong\angle DCE\), this reflection helps in matching the orientation of the non - congruent sides and angles for similarity.

Step 3: Dilation

Since \(AC = 30\) and \(EC=10\), the scale factor of the dilation is \(\frac{EC}{AC}=\frac{10}{30}=\frac{1}{3}\). A dilation with a scale factor of \(\frac{1}{3}\) (centered at point \(C\)) will map \(\triangle ABC\) to \(\triangle EDC\) because when two angles of one triangle are congruent to two angles of another triangle (\(\angle A\cong\angle E\) and \(\angle BCA\cong\angle DCE\)), and the sides around the non - congruent angles are in proportion (after dilation), the triangles are similar by the AA (angle - angle) similarity criterion.

Answer:

Step 2: Reflection across \(\overline{AC}\)
Step 3: Dilation with a scale factor of \(\frac{1}{3}\) (centered at point \(C\))