Question

drag each reason to the correct location on the flow chart. not all reasons will be used. 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. asa reflexive prop of congruence sas alternate inter angles theor cpctc alternate exter angles theor \\( \overline { a c } \cong \overline { a c } \\)

Explanation:

Step1: Justify $\overline{AC}\cong\overline{AC}$

By the reflexive property of congruence, any segment is congruent to itself. So, the reason for $\overline{AC}\cong\overline{AC}$ is "reflexive property of congruence".

Step2: Justify $\angle DAC\cong\angle BCA$ and $\angle ACD\cong\angle BAC$

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

Step3: Justify $\triangle ACD\cong\triangle CAB$

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 for $\triangle ACD\cong\triangle CAB$ is "ASA".

Step4: Justify $\overline{AB}\cong\overline{CD}$ 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 $\overline{AD}\cong\overline{BC}$ by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). So, the reason for $\overline{AB}\cong\overline{CD}$ and $\overline{AD}\cong\overline{BC}$ is "CPCTC".

Answer:

• For $\overline{AC}\cong\overline{AC}$: reflexive property of congruence
• For $\angle DAC\cong\angle BCA$ and $\angle ACD\cong\angle BAC$: alternate interior angles theorem
• For $\triangle ACD\cong\triangle CAB$: ASA
• For $\overline{AB}\cong\overline{CD}$ and $\overline{AD}\cong\overline{BC}$: CPCTC