Question

triangle congruence: sas\
given: c is the midpoint of \\(\overline{bd}\\).\
prove: \\(\triangle acb \cong \triangle acd\\)\
complete the two - column proof.\
\

statements	reasons	\
---	---	\
1. c is the midpoint of \\(\overline{bd}\\)	1. given	\
2. \\(\overline{bc} \cong \overline{cd}\\)	2. \\(\boldsymbol{\downarrow}\\)	\
3. \\(\overline{ac} \cong \overline{ac}\\)	3. reflexive property	\
4. \\(\angle bca\\) and \\(\angle dca\\) are right \\(\angle\\)s	4. given	\
5. \\(\boldsymbol{\downarrow}\\)	5. all right \\(\angle\\)s are \\(\cong\\)	\
6. \\(\triangle acb \cong \triangle acd\\)	6. sas

Explanation:

Step1: Analyze Statement 2

Since \( C \) is the midpoint of \( \overline{BD} \), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So \( BC = CD \), and the reason for \( \overline{BC} \cong \overline{CD} \) is "definition of midpoint".

Step2: Analyze Statement 5

We know \( \angle BCA \) and \( \angle DCA \) are right angles (from statement 4, given), and we need to state the congruence of these angles. The reason "all right angles are congruent" is used, so the statement for 5 should be \( \angle BCA \cong \angle DCA \) (since they are right angles and all right angles are congruent).

Answer:

• For Statement 2 (Reason): definition of midpoint
• For Statement 5 (Statement): \( \angle BCA \cong \angle DCA \)