Question

prove: \\( \overline { a b } \parallel \overline { c d } \\) and \\( \overline { a d } \parallel \overline { b c } \\)
complete the flowchart proof.
\\( \

$$\begin{array} { l } { \\text { given } } \\\\ { \\begin{array} { l } { \\overline { a o } \\cong \\overline { o c } } \\\\ { \\overline { b o } \\cong \\overline { o d } } \\end{array}$$

} \\\\ { \

$$\begin{array} { l } { \\angle a o d \\cong \\angle c o b } \\\\ { \\angle a o b \\cong \\angle c o d } \\end{array}$$

} \\\\ { \

$$\begin{array} { l } { \\triangle a o d \\cong \\triangle c o b } \\\\ { \\triangle a o b \\cong \\triangle c o d } \\end{array}$$

} \\\\ { \

$$\begin{array} { l } { \\angle d a c \\cong \\angle b c a } \\\\ { \\angle b a c \\cong \\angle d c a } \\end{array}$$

} \\\\ { \

$$\begin{array} { l } { \\overline { a b } \\parallel \\overline { c d } } \\\\ { \\overline { a d } \\parallel \\overline { b c } } \\end{array}$$

} \end{array} \\)

Explanation:

Step1: Identify the congruent triangles

We have \(AO = OC\), \(BO=OD\), and \(\angle AOB=\angle COD\) (vertical angles theorem). By the \(SAS\) (Side - Angle - Side) congruence criterion, \(\triangle AOB\cong\triangle COD\).

Step2: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle AOB\cong\triangle COD\), then \(\angle BAC=\angle DCA\).

Step3: Apply the converse of the alternate interior angles theorem

If \(\angle BAC=\angle DCA\), then \(AB\parallel CD\) (converse of alternate interior angles theorem).

Also, for \(\triangle AOD\) and \(\triangle COB\), we have \(AO = OC\), \(BO = OD\), and \(\angle AOD=\angle COB\) (vertical angles theorem). By \(SAS\) congruence, \(\triangle AOD\cong\triangle COB\).

Step4: Use CPCTC again

Since \(\triangle AOD\cong\triangle COB\), then \(\angle DAC=\angle BCA\).

Step5: Apply the converse of the alternate interior angles theorem

If \(\angle DAC=\angle BCA\), then \(AD\parallel BC\) (converse of alternate interior angles theorem).

Answer:

From top to bottom: \(SAS\), \(CPCTC\), \(converse\ of\ alternate\ interior\ angles\ theorem\)