Question

which sides of quadrilateral abcd must be parallel? explain.
m∠a = 52, m∠b = 33, m∠c = 147, m∠d = 128
the sides cd and ab by the
converse of the corresponding angles postulate
converse of the same - side interior angles theorem
converse of the alternate interior angles theorem
converse of the alternate exterior angles theorem

Explanation:

Step1: Check angle sum for parallel - side condition

We know that if two lines are cut by a transversal, and the sum of same - side interior angles is \(180^{\circ}\), then the two lines are parallel.
For sides \(AD\) and \(BC\):
\(m\angle A+m\angle B = 52 + 33=85
eq180\)
\(m\angle C+m\angle D=147 + 128 = 275
eq180\)
For sides \(AB\) and \(CD\):
\(m\angle A+m\angle D=52 + 128=180\)
\(m\angle B+m\angle C=33 + 147 = 180\)

Step2: Apply the theorem

Since \(m\angle A+m\angle D = 180\) and \(m\angle B+m\angle C=180\), by the converse of the same - side interior angles theorem (If two lines are cut by a transversal and the sum of a pair of same - side interior angles is \(180^{\circ}\), then the two lines are parallel), sides \(AB\) and \(CD\) are parallel.

Answer:

converse of the same - side interior angles theorem