Question

9
given: \\( \triangle a b d \cong \triangle c b e \\)
prove: \\( \triangle a e d \cong \triangle c d e \\)
proof:
which of the following reasons completes the proof?
o a. angle-side-angle
o b. angle-angle-side
o c. side-side-side
o d. side-angle-side

Explanation:

Step1: Recall triangle congruence theorems

• Side - Side - Side (SSS): If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
• Side - Angle - Side (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
• Angle - Side - Angle (ASA): 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.
• Angle - Angle - Side (AAS): If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.

Step2: Identify the congruent parts for \(\triangle AED\) and \(\triangle CDE\)

• From the proof:
• \(\overline{AD}\cong\overline{CE}\) (from step 2, \( \triangle ABD\cong\triangle CBE\))
• \(\overline{CD}\cong\overline{AE}\) (from step 7)
• \(\overline{ED}\cong\overline{ED}\) (common side)

Step3: Apply the SSS theorem

Since \( \overline{AD}\cong\overline{CE}\), \( \overline{CD}\cong\overline{AE}\), and \( \overline{ED}\cong\overline{ED}\), by the Side - Side - Side (SSS) congruence theorem, \( \triangle AED\cong\triangle CDE\)

Answer:

C. Side - Side - Side