Question

practice complete the two - column proof by dragging the statements into the correct order. given: ( ba cong cd ), ( overline{ab} parallel overline{cd} ) prove: ( \triangle acd cong \triangle dba ) proof: (\begin{array}{|l|l|}hline \text{statements} & \text{reasons} \\ hline \text{1.?} & \text{a. given} \\ hline \text{2.?} & \text{b. alternate interior angles thm.} \\ hline \text{3.?} & \text{c. reflexive property of } cong \\ hline \text{4.?} & \text{d. def. of congruent segments} \\ hline \text{5.?} & \text{e. sas} \\ hline end{array}) (\boxed{1}) keyboard help ( ba = cd, overline{ab} parallel overline{cd} ) ( overline{ad} cong overline{ad} ) ( \triangle acd cong \triangle dba ) ( angle bad cong angle cda ) ( ba cong cd )

Explanation:

Step1: Analyze Given Information

We know we need to prove \(\triangle ACD \cong \triangle DBA\) with given \(BA = CD\) and \(AB \parallel CD\). Let's match each statement to its reason.

Step2: Match \(BA = CD\)

The statement \(BA = CD\) is given, so its reason is "A. Given". So the statement \(BA \cong CD\) (since \(BA = CD\) implies congruent segments) has reason "A. Def. of congruent segments"? Wait, no, \(BA = CD\) is given, so first, the statement \(BA = CD\) (or \(BA \cong CD\)): the reason for \(BA \cong CD\) when \(BA = CD\) is "A. Def. of congruent segments" (since congruent segments have equal length). Wait, let's list the statements and reasons:

1. Statement: \(BA = CD, AB \parallel CD\) → Reason: "A. Given" (since these are the given facts).
2. Statement: \(\angle BAD \cong \angle CDA\) → Reason: "E. Alternate Interior Angles Thm." (because \(AB \parallel CD\) and transversal \(AD\), so alternate interior angles are congruent).
3. Statement: \(AD \cong AD\) → Reason: "C. Reflexive Property of \(\cong\)" (a segment is congruent to itself).
4. Statement: \(BA \cong CD\) → Reason: "A. Def. of congruent segments" (since \(BA = CD\) is given, by definition of congruent segments, they are congruent).
5. Statement: \(\triangle ACD \cong \triangle DBA\) → Reason: "B. SAS" (we have \(BA \cong CD\), \(\angle BAD \cong \angle CDA\), \(AD \cong AD\), so SAS congruence).

Now let's match the boxes:

- The first box: \(BA = CD, AB \parallel CD\) → Reason "A. Given" (so this statement goes with reason A).
- The second box: \(AD \cong AD\) → Reason "C. Reflexive Property of \(\cong\)".
- The third box: \(\triangle ACD \cong \triangle DBA\) → Reason "B. SAS".
- The fourth box: \(\angle BAD \cong \angle CDA\) → Reason "E. Alternate Interior Angles Thm.".
- The fifth box: \(BA \cong CD\) → Reason "A. Def. of congruent segments" (wait, maybe the statements are:

Wait the statements in the lower boxes are:

1. \(BA = CD, AB \parallel CD\)
2. \(AD \cong AD\)
3. \(\triangle ACD \cong \triangle DBA\)
4. \(\angle BAD \cong \angle CDA\)
5. \(BA \cong CD\)

And the reasons are:

A. Given

E. Alternate Interior Angles Thm.

C. Reflexive Property of \(\cong\)

A. Def. of congruent segments (wait, maybe the first "A" is "Given" and the other "A" is "Def. of congruent segments"? Wait the problem has reasons: A. Given, E. Alternate Interior Angles Thm., C. Reflexive Property of \(\cong\), A. Def. of congruent segments, B. SAS.

So let's assign:

- Statement \(BA = CD, AB \parallel CD\) → Reason A (Given)
- Statement \(\angle BAD \cong \angle CDA\) → Reason E (Alternate Interior Angles Thm., because \(AB \parallel CD\), transversal \(AD\))
- Statement \(AD \cong AD\) → Reason C (Reflexive Property)
- Statement \(BA \cong CD\) → Reason A (Def. of congruent segments, since \(BA = CD\))
- Statement \(\triangle ACD \cong \triangle DBA\) → Reason B (SAS: \(BA \cong CD\), \(\angle BAD \cong \angle CDA\), \(AD \cong AD\))

So the matching is:

1. \(BA = CD, AB \parallel CD\) → A (Given)
2. \(\angle BAD \cong \angle CDA\) → E (Alternate Interior Angles Thm.)
3. \(AD \cong AD\) → C (Reflexive Property)
4. \(BA \cong CD\) → A (Def. of congruent segments)
5. \(\triangle ACD \cong \triangle DBA\) → B (SAS)

Now let's put them in order:

- First box (top): \(BA = CD, AB \parallel CD\) → Reason A (Given)
- Second box: \(AD \cong AD\) → Reason C (Reflexive Property)
- Third box: \(\triangle ACD \cong \triangle DBA\) → Reason B (SAS)
- Fourth box: \(\angle BAD \cong \angle CDA\) → Reason E (Alternate Interior Angles Thm.)
- Fifth box: \(BA \cong CD\) → Reason A (Def. of congruent segments)

Wait, maybe the order of statements is:

To prove \(\triangle ACD \cong \triangle DBA\) using SAS, we need:

- Two sides and included angle.

Given \(AB \parallel CD\), so \(\angle BAD = \angle CDA\) (alternate interior angles).

\(BA = CD\) (given), so \(BA \cong CD\) (def of congruent segments).

\(AD\) is common, so \(AD \cong AD\) (reflexive).

Thus, SAS: \(BA \cong CD\), \(\angle BAD \cong \angle CDA\), \(AD \cong AD\), so \(\triangle ACD \cong \triangle DBA\).

So the statements in order:

1. \(BA = CD, AB \parallel CD\) (Given)
2. \(\angle BAD \cong \angle CDA\) (Alternate Interior Angles Thm.)
3. \(BA \cong CD\) (Def. of congruent segments, since \(BA = CD\))
4. \(AD \cong AD\) (Reflexive Property)
5. \(\triangle ACD \cong \triangle DBA\) (SAS)

So matching to the boxes:

- First box: \(BA = CD, AB \parallel CD\) → Reason A (Given)
- Second box: \(AD \cong AD\) → Reason C (Reflexive)
- Third box: \(\triangle ACD \cong \triangle DBA\) → Reason B (SAS)
- Fourth box: \(\angle BAD \cong \angle CDA\) → Reason E (Alternate Interior Angles Thm.)
- Fifth box: \(BA \cong CD\) → Reason A (Def. of congruent segments)

Now let's place each statement in the correct box:

1. The first box (top) is for \(BA = CD, AB \parallel CD\) (Given) → Reason A.
2. The second box: \(AD \cong AD\) → Reason C.
3. The third box: \(\triangle ACD \cong \triangle DBA\) → Reason B.
4. The fourth box: \(\angle BAD \cong \angle CDA\) → Reason E.
5. The fifth box: \(BA \cong CD\) → Reason A (Def. of congruent segments).

So the correct matching is:

- \(BA = CD, AB \parallel CD\) → A (Given)
- \(\angle BAD \cong \angle CDA\) → E (Alternate Interior Angles Thm.)
- \(BA \cong CD\) → A (Def. of congruent segments)
- \(AD \cong AD\) → C (Reflexive Property)
- \(\triangle ACD \cong \triangle DBA\) → B (SAS)

Now let's assign each box:

1. First box (top): \(BA = CD, AB \parallel CD\) → Reason A (Given)
2. Second box: \(AD \cong AD\) → Reason C (Reflexive)
3. Third box: \(\triangle ACD \cong \triangle DBA\) → Reason B (SAS)
4. Fourth box: \(\angle BAD \cong \angle CDA\) → Reason E (Alternate Interior Angles Thm.)
5. Fifth box: \(BA \cong CD\) → Reason A (Def. of congruent segments)

So the order of statements (from top to bottom boxes) is:

1. \(BA = CD, AB \parallel CD\)
2. \(\angle BAD \cong \angle CDA\)
3. \(BA \cong CD\)
4. \(AD \cong AD\)
5. \(\triangle ACD \cong \triangle DBA\)

Wait, no, the boxes are:

Top box: \(BA = CD, AB \parallel CD\)

Next: \(AD \cong AD\)

Next: \(\triangle ACD \cong \triangle DBA\)

Next: \(\angle BAD \cong \angle CDA\)

Next: \(BA \cong CD\)

So let's match each to reason:

- \(BA = CD, AB \parallel CD\) → A (Given)
- \(AD \cong AD\) → C (Reflexive)
- \(\triangle ACD \cong \triangle DBA\) → B (SAS)
- \(\angle BAD \cong \angle CDA\) → E (Alternate Interior Angles Thm.)
- \(BA \cong CD\) → A (Def. of congruent segments)

Yes, that makes sense. So the correct matching is:

1. \(BA = CD, AB \parallel CD\) → A (Given)
2. \(AD \cong AD\) → C (Reflexive Property)
3. \(\triangle ACD \cong \triangle DBA\) → B (SAS)
4. \(\angle BAD \cong \angle CDA\) → E (Alternate Interior Angles Thm.)
5. \(BA \cong CD\) → A (Def. of congruent segments)

Answer:

1. \(BA = CD, AB \parallel CD\) with Reason A (Given)
2. \(AD \cong AD\) with Reason C (Reflexive Property of \(\cong\))
3. \(\triangle ACD \cong \triangle DBA\) with Reason B (SAS)
4. \(\angle BAD \cong \angle CDA\) with Reason E (Alternate Interior Angles Thm.)
5. \(BA \cong CD\) with Reason A (Def. of congruent segments)

(Note: The exact placement depends on the box order, but the key is matching each statement to its reason as above.)