Question

a. what is reason 1?
given ad||bc
prove ( m angle 1 + m angle 2 + m angle 3 = 180 ^ { circ } ).

Explanation:

Step1: Recall the property of congruent angles

If \(\angle1=\angle4\), by the definition of congruent angles (which states that if two angles are congruent, then their measures are equal), we have \(m\angle1 = m\angle4\). This is the reason for the statement \(m\angle1=m\angle4\) (line 2 in the table).

Step2: Recall the property of parallel lines

When \(AD\parallel BC\), we use the property that if two parallel lines are cut by a transversal, then alternate - interior angles are congruent. Here, \(\angle2\) and \(\angle5\) are alternate - interior angles. So, by the definition of congruent angles (if \(\angle2\cong\angle5\), then \(m\angle2 = m\angle5\)), which is the reason for the statement \(m\angle2=m\angle5\) (line 3 in the table).

Step3: Recall the angle - addition postulate

The angle - addition postulate states that if we have angles that form a larger angle, the sum of the measures of the smaller angles is equal to the measure of the larger angle. For \(\angle EAD\), if \(\angle4+\angle3+\angle5\) form \(\angle EAD\), then \(m\angle4 + m\angle3+m\angle5=m\angle EAD\) (line 4 in the table).

Step4: Recall the property of supplementary angles

When two angles form a linear pair (a straight line), they are supplementary. If \(AD\parallel BC\) and considering the transversal that forms \(\angle4,\angle3,\angle5\) such that they form a linear - pair - like situation (in the context of the parallel - line and transversal setup), \(m\angle4 + m\angle3+m\angle5 = 180^{\circ}\) (line 5 in the table).

Step5: Recall the substitution property

Since \(m\angle1 = m\angle4\) and \(m\angle2=m\angle5\), we substitute \(m\angle1\) for \(m\angle4\) and \(m\angle2\) for \(m\angle5\) in the equation \(m\angle4 + m\angle3+m\angle5 = 180^{\circ}\). So, \(m\angle1 + m\angle2+m\angle3 = 180^{\circ}\) (line 6 in the table). The first reason (\(m\angle1 = m\angle4\) from \(\angle1=\angle4\)) is based on the Definition of congruent angles (if \(\angle A\cong\angle B\), then \(m\angle A=m\angle B\)).

Answer:

Definition of congruent angles.