Question

use the diagram. given: ∠a and ∠c are right angles; \\(\overline{ab} \cong \overline{cd}\\). prove: \\(\triangle abd \cong \triangle cdb\\) complete the proof. statements reasons 1. ∠a and ∠c are right angles; \\(\overline{ab} \cong \overline{cd}\\). 1. 2. \\(\triangle abd\\) and \\(\triangle cdb\\) are right triangles. 2. 3. \\(\overline{bd} \cong \overline{db}\\) 3. 4. \\(\triangle abd \cong \triangle cdb\\) 4. drag the reasons to the correct order

Explanation:

Step1: Identify Given Info

Given \( \angle A, \angle C \) are right angles (\( 90^\circ \)) and \( \overline{AB} \cong \overline{CD} \). Reason: Given.

Step2: Identify Common Side

\( \overline{BD} \cong \overline{DB} \) (reflexive property of congruence, a side is congruent to itself). Reason: Reflexive Property of Congruence.

Step3: Classify Triangles

\( \triangle ABD \) and \( \triangle CDB \) are right triangles (since \( \angle A, \angle C \) are right angles). Reason: Definition of Right Triangle (a triangle with one right angle).

Step4: Prove Congruence

\( \triangle ABD \cong \triangle CDB \) by HL (Hypotenuse - Leg) Congruence Theorem (right triangles with hypotenuse \( BD \) and leg \( AB \cong CD \)). Reason: HL Congruence Theorem.

Answer:

1. Reason: Given
2. Reason: Definition of Right Triangle
3. Reason: Reflexive Property of Congruence
4. Reason: HL Congruence Theorem

(For the table, fill each "Reasons" cell with the above respective reasons.)