Question

matching 33 points
proof complete the proof.
given: \\(\overline{ab} \parallel \overline{cd}\\), \\(\overline{ab} \cong \overline{cd}\\)
prove: \\(\triangle abc \cong \triangle dcb\\)

statements | reasons
--- | ---

1. \\(\overline{ab} \parallel \overline{cd}\\) | 1. ?
2. \\(\angle abc \cong \angle dcb\\) | 2. ?
3. \\(\overline{ab} \cong \overline{cd}\\) | 3. ?
4. \\(\overline{cb} \cong \overline{cb}\\) | 4. ?
5. \\(\triangle abc \cong \triangle dcb\\) | 5. ?

statement 1 |
statement 2 | given
statement 3 | sas
statement 4 | aas
statement 5 | alternate interior angles
segment ac is congruent to segment bd | angle bisector
| cpctc
| asa
| reflexive property

Explanation:

Step1: Analyze Statement 1

Statement 1 is $\overline{AB} \parallel \overline{CD}$, which is given in the problem. So the reason for Statement 1 is "Given".

Step2: Analyze Statement 2

Since $\overline{AB} \parallel \overline{CD}$ and $\overline{CB}$ is a transversal, $\angle ABC$ and $\angle DCB$ are alternate interior angles. So the reason for $\angle ABC \cong \angle DCB$ is "Alternate Interior Angles".

Step3: Analyze Statement 3

Statement 3 is $\overline{AB} \cong \overline{CD}$, which is given in the problem. So the reason for Statement 3 is "Given".

Step4: Analyze Statement 4

$\overline{CB} \cong \overline{CB}$ because a segment is congruent to itself, which is the Reflexive Property. So the reason for Statement 4 is "Reflexive Property".

Step5: Analyze Statement 5

We have $\overline{AB} \cong \overline{CD}$, $\angle ABC \cong \angle DCB$, and $\overline{CB} \cong \overline{CB}$. This satisfies the SAS (Side - Angle - Side) congruence criterion for triangles. So the reason for $\triangle ABC \cong \triangle DCB$ is "SAS".

Answer:

1. Given
2. Alternate Interior Angles
3. Given
4. Reflexive Property
5. SAS