Question

given: \\( \overline { c d } \cong \overline { b e } , \overline { c d } \parallel \overline { b e } \\), and \\( \overline { b c } \cong \overline { a c } \\) prove: \\( \angle d \cong \angle b \\) arrange the cards to complete the proof. fill in the blank where indicated. mark sets of congruent sides and angles if the corresponding congruence statement is justified in the proof.

Explanation:

Step1: Identify the given side

Given \( \overline{BC}\cong\overline{AC}\), so in row 3, we have \( \overline{BC}\cong\overline{AC}\).

Step2: Find congruent angles

Since \( \overline{CD}\parallel\overline{BE}\), by the alternate - interior angles theorem (which is a case of corresponding angles for parallel lines cut by a transversal), \( \angle ACD\cong\angle BCE\).

Step3: Determine congruent triangles

We have \( \overline{CD}\cong\overline{BE}\) (given), \( \overline{BC}\cong\overline{AC}\) (given), \( \angle ACD\cong\angle BCE\) (from step 2). By the Side - Angle - Side (SAS) triangle congruence theorem, \( \triangle ACD\cong\triangle BCE\).

Answer:

Row 3: \( \overline{BC}\cong\overline{AC}\); Row 4: \( \angle ACD\cong\angle BCE\); Row 5: \( \triangle ACD\cong\triangle BCE\), SAS.