Question

building design the aldar properties headquarters building in the united arab emirates is a unique structural design comprising geometric shapes. if ab ≅ dc and bd ≅ ca, complete the paragraph proof to prove that △abd ≅ △dca. proof: we know that ab select choice to dc and bd select choice to ca. by the select choice property of congruence we know ad ≅ da. therefore △abd ≅ △dca by select choice.

Explanation:

To solve the proof that \( \triangle ABD \cong \triangle DCA \), we analyze the given information and congruence properties:

Step 1: Identify Congruent Sides (First Blank)

We know \( \overline{AB} \cong \overline{DC} \) (given or from the problem’s context, as \( AB \) and \( DC \) are sides of the quadrilateral).

Step 2: Identify Congruent Sides (Second Blank)

We know \( \overline{BD} \cong \overline{CA} \) (given in the problem: \( BD \cong CA \)).

Step 3: Identify the Reflexive Property (Third Blank)

The segment \( \overline{AD} \) is common to both \( \triangle ABD \) and \( \triangle DCA \). By the Reflexive Property of Congruence, \( \overline{AD} \cong \overline{DA} \) (a segment is congruent to itself).

Step 4: Determine the Congruence Criterion (Fourth Blank)

Now, we have three pairs of congruent sides:

• \( \overline{AB} \cong \overline{DC} \)
• \( \overline{BD} \cong \overline{CA} \)
• \( \overline{AD} \cong \overline{DA} \)

By the SSS (Side - Side - Side) Congruence Criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Thus, \( \triangle ABD \cong \triangle DCA \) by SSS.

Final Answers (Filling the Blanks):

1. First blank: \( \cong \) (for \( \overline{AB} \cong \overline{DC} \))
2. Second blank: \( \cong \) (for \( \overline{BD} \cong \overline{CA} \))
3. Third blank: Reflexive
4. Fourth blank: SSS

(Note: If the blanks are dropdowns with options like “is congruent to”, “Reflexive”, “SSS”, the answers align with these choices.)

Answer:

To solve the proof that \( \triangle ABD \cong \triangle DCA \), we analyze the given information and congruence properties:

Step 1: Identify Congruent Sides (First Blank)

We know \( \overline{AB} \cong \overline{DC} \) (given or from the problem’s context, as \( AB \) and \( DC \) are sides of the quadrilateral).

Step 2: Identify Congruent Sides (Second Blank)

We know \( \overline{BD} \cong \overline{CA} \) (given in the problem: \( BD \cong CA \)).

Step 3: Identify the Reflexive Property (Third Blank)

The segment \( \overline{AD} \) is common to both \( \triangle ABD \) and \( \triangle DCA \). By the Reflexive Property of Congruence, \( \overline{AD} \cong \overline{DA} \) (a segment is congruent to itself).

Step 4: Determine the Congruence Criterion (Fourth Blank)

Now, we have three pairs of congruent sides:

• \( \overline{AB} \cong \overline{DC} \)
• \( \overline{BD} \cong \overline{CA} \)
• \( \overline{AD} \cong \overline{DA} \)

By the SSS (Side - Side - Side) Congruence Criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Thus, \( \triangle ABD \cong \triangle DCA \) by SSS.

Final Answers (Filling the Blanks):

1. First blank: \( \cong \) (for \( \overline{AB} \cong \overline{DC} \))
2. Second blank: \( \cong \) (for \( \overline{BD} \cong \overline{CA} \))
3. Third blank: Reflexive
4. Fourth blank: SSS

(Note: If the blanks are dropdowns with options like “is congruent to”, “Reflexive”, “SSS”, the answers align with these choices.)