Question

given: abcd is a parallelogram.
prove: ( mangle a + mangle b + mangle c + mangle d = 360^{circ})
by the definition of a parallelogram, (overline{ad}paralleloverline{bc}) and (overline{ab}paralleloverline{dc}). using, (overline{ad}) as a transversal, (angle a) and (angle d) are same - side interior angles, so they are supplementary. by the definition of supplementary, (mangle a + mangle d = 180). using side (overline{bc}) as a transversal, (angle b) and (angle c) are same - side interior angles, so they are supplementary. by the definition of supplementary, (mangle b + mangle c = 180). so, (mangle a + mangle d + mangle b + mangle c = 360) by the substitution property. simplifying, we have (mangle a + mangle b + mangle c + mangle d = 360^{circ}).

Explanation:

Step1: Analyze the first pair of same - side interior angles

Given \(AD\parallel BC\) and \(AD\) as a transversal. For \(\angle A\) and \(\angle D\), since \(AD\parallel BC\) and \(AB\) is a transversal (in a parallelogram \(AB\) is a side), but when considering \(AD\) as a transversal for \(AD\parallel BC\), \(\angle A\) and \(\angle D\) are same - side interior angles. By the property of parallel lines (if two parallel lines are cut by a transversal, same - side interior angles are supplementary), \(m\angle A + m\angle D=180^{\circ}\).

Step2: Analyze the second pair of same - side interior angles

Since \(AB\parallel DC\), using \(BC\) as a transversal. \(\angle B\) and \(\angle C\) are same - side interior angles. By the property of parallel lines (if two parallel lines are cut by a transversal, same - side interior angles are supplementary), \(m\angle B + m\angle C = 180^{\circ}\).

Step3: Sum the two equations

We know that \(m\angle A+m\angle D = 180^{\circ}\) and \(m\angle B + m\angle C=180^{\circ}\). Then \(m\angle A+m\angle D+m\angle B + m\angle C=(m\angle A + m\angle D)+(m\angle B + m\angle C)\). Substituting the values from the two equations: \(180^{\circ}+180^{\circ}=360^{\circ}\), so \(m\angle A + m\angle B+m\angle C + m\angle D = 360^{\circ}\).

Answer:

The second blank (the side used as a transversal for \(\angle B\) and \(\angle C\)) should be filled with \(BC\).