Question

complete the proof below.
given: $overline{ad} \cong overline{cb}$, $overline{ab} \cong overline{dc}$
prove: $abcd$ is a parallelogram
statements

1. $overline{ad} \cong overline{cb}$, $overline{ab} \cong overline{dc}$
2. $overline{ac} \cong overline{ac}$
3. $\triangle abc \cong \triangle cda$
4. $angle bac \cong angle dca$, $angle bca \cong angle dac$
5. $overline{ab} \parallel overline{dc}$, $overline{ad} \parallel overline{bc}$

6.
reasons

1. given
2. reflexive property of congruence

3.
4.
5.
6.
also, (4x)°, and m∠d = (2x + 10)°.

Explanation:

Step1: Analyze Given Information

We know \( \overline{AD} \cong \overline{CB} \), \( \overline{AB} \cong \overline{DC} \), and \( \overline{AC} \cong \overline{AC} \) (reflexive property). So by SSS (Side - Side - Side) congruence criterion, \( \triangle ABC \cong \triangle CDA \).

Step2: Use Congruent Triangles to Find Angles

Since \( \triangle ABC \cong \triangle CDA \), corresponding angles are congruent. So \( \angle BAC \cong \angle DCA \) and \( \angle BCA \cong \angle DAC \).

Step3: Determine Parallel Sides

If alternate interior angles are congruent, then the lines are parallel. Since \( \angle BAC \cong \angle DCA \), \( \overline{AB} \parallel \overline{DC} \) (alternate interior angles theorem). Since \( \angle BCA \cong \angle DAC \), \( \overline{AD} \parallel \overline{BC} \) (alternate interior angles theorem).

Step4: Conclude it's a Parallelogram

A quadrilateral with both pairs of opposite sides parallel is a parallelogram. So \( ABCD \) is a parallelogram.

Now, let's fill the proof table:

1. Statement: \( \overline{AD} \cong \overline{CB}, \overline{AB} \cong \overline{DC} \)

Reason: Given

1. Statement: \( \overline{AC} \cong \overline{AC} \)

Reason: Reflexive Property of Congruence

1. Statement: \( \triangle ABC \cong \triangle CDA \)

Reason: SSS (Side - Side - Side) Congruence Postulate

1. Statement: \( \angle BAC \cong \angle DCA, \angle BCA \cong \angle DAC \)

Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

1. Statement: \( \overline{AB} \parallel \overline{DC}, \overline{AD} \parallel \overline{BC} \)

Reason: Alternate Interior Angles Theorem (If alternate interior angles are congruent, then lines are parallel)

1. Statement: \( ABCD \) is a parallelogram

Reason: Definition of a Parallelogram (A quadrilateral with both pairs of opposite sides parallel is a parallelogram)

Answer:

The completed proof table has the following reasons (and statements as above):

1. Given
2. Reflexive Property of Congruence
3. SSS Congruence Postulate
4. CPCTC
5. Alternate Interior Angles Theorem
6. Definition of a Parallelogram

(And the statements are as analyzed in the steps above to show \( ABCD \) is a parallelogram)