Question

given: \\( \overline { a b } \parallel \overline { c d } \\) and \\( \overline { a d } \parallel \overline { b c } \\)
prove: \\( \overline { a b } \cong \overline { c d } \\) and \\( \overline { a d } \cong \overline { b c } \\)
complete the flow chart proof.
reflexive prop
of congruence
asa
cpctc
alternate exter
angles theore
sas
alternate inter
angles theore
\\( \overline { a c } \cong \overline { a c } \\)
construct \\( \overline { a c } \\)
two points
define a line
\\( \angle d a c \cong \angle b c a \\)
\\( \angle a c d \cong \angle b a c \\)
\\( \triangle a c d \cong \triangle c a b \\)
\\( \overline { a b } \cong \overline { c d } \\)
\\( \overline { a d } \cong \overline { b c } \\)
\\( \overline { a b } \parallel \overline { c d } \\)
\\( \overline { a d } \parallel \overline { b c } \\)
given

Explanation:

Step1: Identify the reason for \( \overline{AC}\cong\overline{AC} \)

The reflexive property of congruence states that any segment is congruent to itself. So, the reason for \( \overline{AC}\cong\overline{AC} \) is "reflexive property of congruence".

Step2: Identify the reason for \( \angle DAC\cong\angle BCA \) and \( \angle ACD\cong\angle BAC \)

When two parallel lines are cut by a transversal, alternate - interior angles are congruent. Since \( \overline{AD}\parallel\overline{BC} \) and \( \overline{AC} \) is a transversal, \( \angle DAC\cong\angle BCA \). Also, since \( \overline{AB}\parallel\overline{CD} \) and \( \overline{AC} \) is a transversal, \( \angle ACD\cong\angle BAC \). So, the reason is "alternate interior angles theorem".

Step3: Identify the reason for \( \triangle ACD\cong\triangle CAB \)

We have two angles and the included side. \( \angle DAC\cong\angle BCA \), \( \overline{AC}\cong\overline{AC} \), \( \angle ACD\cong\angle BAC \). By the ASA (Angle - Side - Angle) congruence criterion, \( \triangle ACD\cong\triangle CAB \). So, the reason is "ASA".

Step4: Identify the reason for \( \overline{AB}\cong\overline{CD} \) and \( \overline{AD}\cong\overline{BC} \)

When two triangles are congruent, their corresponding parts are congruent. Since \( \triangle ACD\cong\triangle CAB \), then \( \overline{AB}\cong\overline{CD} \) and \( \overline{AD}\cong\overline{BC} \). So, the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

1. For \( \overline{AC}\cong\overline{AC} \): reflexive property of congruence
2. For \( \angle DAC\cong\angle BCA \) and \( \angle ACD\cong\angle BAC \): alternate interior angles theorem
3. For \( \triangle ACD\cong\triangle CAB \): ASA
4. For \( \overline{AB}\cong\overline{CD} \) and \( \overline{AD}\cong\overline{BC} \): CPCTC