Drawing the diagonal [tex]\overline {AC}[/tex] of the parallelogram makes the proof of the
congruency of opposite sides of a parallelogram clearer.
[tex]\overline {AB}[/tex] ≅ [tex]\overline {CD}[/tex] and [tex]\overline {BC}[/tex] ≅ [tex]\overline {DA}[/tex] by ASA rule of congruency
Reasons:
The two column proof is presented as follows;
Statement [tex]{}[/tex] Reasons
ABCD is a parallelogram [tex]{}[/tex] Given
∠BAC ≅ ∠DCA [tex]{}[/tex] Alternate interior angles theorem
∠BCA ≅ ∠DAC [tex]{}[/tex] Alternate interior angles theorem
[tex]\overline {AC}[/tex] ≅ [tex]\overline {AC}[/tex] [tex]{}[/tex] Reflexive property
ΔABC ≅ ΔACD [tex]{}[/tex] ASA rule of congruency
[tex]\overline {AB}[/tex] ≅ [tex]\overline {CD}[/tex] [tex]{}[/tex] CPCTC
[tex]\overline {BC}[/tex] ≅ [tex]\overline {DA}[/tex] [tex]{}[/tex] CPCTC
Therefore, segment [tex]\overline {AB}[/tex] and segment [tex]\overline {CB}[/tex] in ΔABC are congruent to
segment [tex]\overline {CD}[/tex] and segment [tex]\overline {DA}[/tex] in ΔACD by Angle-Side-Angle, ASA, rule of
congruency.
CPCTC stands for Congruent Part of Congruent Triangle are Congruent
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