Question

1. given \\(\overline{bc} \cong \overline{dc}\\) and \\(\overline{ac} \cong \overline{ec}\\). prove \\(\triangle bca \cong \triangle dce\\)

1. given: e is the midpoint of \\(\overline{ac}\\)

e is the midpoint of \\(\overline{bd}\\)
prove: \\(\triangle abe \cong \triangle cde\\)

Explanation:

Problem 4

Step1: Identify Vertical Angles

$\angle BCA$ and $\angle DCE$ are vertical angles, so $\angle BCA \cong \angle DCE$ (Vertical Angles Theorem).

Step2: List Given Congruent Sides

Given $\overline{BC} \cong \overline{DC}$ and $\overline{AC} \cong \overline{EC}$.

Step3: Apply SAS Congruence

In $\triangle BCA$ and $\triangle DCE$:

• $\overline{BC} \cong \overline{DC}$ (Given)
• $\angle BCA \cong \angle DCE$ (Vertical Angles)
• $\overline{AC} \cong \overline{EC}$ (Given)

By the Side - Angle - Side (SAS) Congruence Postulate, $\triangle BCA \cong \triangle DCE$.

Step1: Use Midpoint Definition

Since $E$ is the midpoint of $\overline{AC}$, then $\overline{AE} \cong \overline{CE}$ (Definition of Midpoint: a midpoint divides a segment into two congruent segments).
Since $E$ is the midpoint of $\overline{BD}$, then $\overline{BE} \cong \overline{DE}$ (Definition of Midpoint).

Step2: Identify Vertical Angles

$\angle AEB$ and $\angle CED$ are vertical angles, so $\angle AEB \cong \angle CED$ (Vertical Angles Theorem).

Step3: Apply SAS Congruence

In $\triangle ABE$ and $\triangle CDE$:

• $\overline{AE} \cong \overline{CE}$ (From midpoint of $AC$)
• $\angle AEB \cong \angle CED$ (Vertical Angles)
• $\overline{BE} \cong \overline{DE}$ (From midpoint of $BD$)

By the Side - Angle - Side (SAS) Congruence Postulate, $\triangle ABE \cong \triangle CDE$.

Answer:

$\triangle BCA \cong \triangle DCE$ (proven by SAS)

Problem 5