Question

given: c is the midpoint of (overline{bd}). prove: (\triangle acb cong \triangle acd)
diagram: triangle with vertex a, base bd, c as midpoint of bd, right angles at c for (angle bca) and (angle dca)
complete the two - column proof.
statements | reasons

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

Explanation:

Step1: Reason for \( \overline{BC} \cong \overline{CD} \)

The definition of a midpoint states that a midpoint of a segment divides the segment into two congruent segments. Since \( C \) is the midpoint of \( \overline{BD} \), by the definition of midpoint, \( \overline{BC} \cong \overline{CD} \).

Step2: Statement for \( \angle BCA \cong \angle DCA \)

We know that \( \angle BCA \) and \( \angle DCA \) are right angles (from step 4). The reason given for step 5 is "all right \( \angle \)'s are \( \cong \)", so the statement should be \( \angle BCA \cong \angle DCA \).

Answer:

• For the first blank (reason for \( \overline{BC} \cong \overline{CD} \)): definition of midpoint
• For the second blank (statement for step 5): \( \angle BCA \cong \angle DCA \)