Question

1. write a two - column proof. given: \\(\overline{ac}\\) bisects \\(\angle bad\\), \\(\angle bca\cong\angle dca\\) prove: \\(bc = dc\\) \\(\

$$\begin{array}{|c|c|c|}\\hline\\text{statements}&\\text{reasons}\\\\\\hline1.&1.\\\\\\hline2.&2.\\\\\\hline3.&3.\\\\\\hline4.&4.\\\\\\hline5.&5.\\\\\\hline6.&6.\\\\\\hline7.&7.\\\\\\hline\\end{array}$$

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Explanation:

Step1: Identify Given Information

• $\overline{AC}$ bisects $\angle BAD$: So $\angle BAC \cong \angle DAC$ (Definition of Angle Bisector).
• $\angle BCA \cong \angle DCA$ (Given).
• $\overline{AC} \cong \overline{AC}$ (Reflexive Property of Congruence).

Step2: Prove Triangles Congruent

By the ASA (Angle - Side - Angle) Congruence Postulate, $\triangle BCA \cong \triangle DCA$ because:

• $\angle BAC \cong \angle DAC$ (from step 1),
• $\overline{AC} \cong \overline{AC}$ (from step 1),
• $\angle BCA \cong \angle DCA$ (given).

Step3: Corresponding Parts of Congruent Triangles

Since $\triangle BCA \cong \triangle DCA$, their corresponding sides are congruent. So $\overline{BC} \cong \overline{DC}$, which means $BC = DC$ (Definition of Congruent Segments).

Now, we can fill the two - column proof:

Statements	Reasons
2. $\angle BAC \cong \angle DAC$	Definition of Angle Bisector
3. $\angle BCA \cong \angle DCA$	Given
4. $\overline{AC} \cong \overline{AC}$	Reflexive Property of Congruence
5. $\triangle BCA \cong \triangle DCA$	ASA Congruence Postulate (Steps 2, 4, 3)
6. $BC = DC$	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Answer:

The two - column proof is filled as shown above, and we conclude that $BC = DC$ by proving $\triangle BCA \cong \triangle DCA$ using ASA and then applying CPCTC.