Question

#2 proving lines parallel
name
given: ∠abc≅∠adc
m∠a + m∠abc = 180
prove: ab∥cd
statements
∠abc≅∠adc ; m∠a + m∠abc = 180
m∠abc = m∠adc
m∠a + m∠adc = 180
∠a and ∠adc are supplementary
ab∥cd
reasons

Explanation:

Step1: Given information

Given $\angle ABC\cong\angle ADC$ and $m\angle A + m\angle ABC=180$.

Step2: Use congruence property

Since $\angle ABC\cong\angle ADC$, by the definition of congruent angles, $m\angle ABC = m\angle ADC$.

Step3: Substitute

Substitute $m\angle ABC$ with $m\angle ADC$ in $m\angle A + m\angle ABC = 180$, we get $m\angle A + m\angle ADC=180$.

Step4: Define supplementary angles

Two angles are supplementary if the sum of their measures is 180. So $\angle A$ and $\angle ADC$ are supplementary.

Step5: Use parallel - line theorem

If two lines are cut by a transversal and a pair of same - side interior angles are supplementary, then the two lines are parallel. Here, $\overline{AB}$ and $\overline{CD}$ are cut by transversal $\overline{AD}$, and $\angle A$ and $\angle ADC$ are same - side interior angles. So $\overline{AB}\parallel\overline{CD}$.

Answer:

Statements	Reasons
$m\angle ABC = m\angle ADC$	Definition of congruent angles
$m\angle A + m\angle ADC=180$	Substitution property of equality
$\angle A$ and $\angle ADC$ are supplementary	Definition of supplementary angles
$\overline{AB}\parallel\overline{CD}$	Same - side interior angles postulate