Question

given: \\( \triangle a b c ; \angle a \cong \angle b \\) prove: \\( \overline{a c} \cong \overline{b c} \\) statements reasons 1. \\( \triangle a b c ; \angle a \cong \angle b \\) 1. given 2. draw auxiliary line \\( \overline{c m} \\) bisecting \\( \angle c \\) 2. every angle has a unique bisector 3. \\( \angle a c m \cong \angle b c m \\) 3. def \\( \angle \\) bisector 4. \\( \overline{c m} \cong \overline{c m} \\) 4. prop. of \\( \cong \\) 5. \\( \triangle a c m \cong \triangle b c m \\) 5. 6. 6. cpctc a. \\( \overline{a m} \cong \overline{m b} \\) b. \\( \angle a m c \cong \angle b m c \\) c. aas d. sas e. reflexive f. \\( \overline{a c} \cong \overline{b c} \\)

Explanation:

Step 1: Reason for $\overline{CM}\cong\overline{CM}$

The reflexive property of congruence states that any segment is congruent to itself. So, for reason 4, the answer is "Reflexive".

Step 2: Congruence criterion for $\triangle ACM\cong\triangle BCM$

We have $\angle A\cong\angle B$ (given), $\angle ACM\cong\angle BCM$ (by definition of angle - bisector), and $\overline{CM}\cong\overline{CM}$ (reflexive property). This satisfies the AAS (Angle - Angle - Side) congruence criterion. So, for reason 5, the answer is "AAS".

Step 3: Conclusion using CPCTC

Since we want to prove $\overline{AC}\cong\overline{BC}$, and by CPCTC (Corresponding Parts of Congruent Triangles are Congruent) from $\triangle ACM\cong\triangle BCM$, for statement 6, the answer is $\overline{AC}\cong\overline{BC}$.

Answer:

1. e. Reflexive
2. c. AAS
3. f. $\overline{AC}\cong\overline{BC}$