Question

can the following quadrilateral be proven to be a parallelogram based on the given information? explain.

choose the correct answer below.
a. yes. it is a parallelogram because both pairs of opposite angles are congruent.
b. no. it is not a parallelogram because one pair of opposite sides is not congruent and the other pair of opposite sides is congruent.
c. no. it is not a parallelogram because the angles of the quadrilateral do not add up to 360°.
d. yes. it is a parallelogram because one pair of opposite sides is congruent and the other pair of opposite sides is parallel.

Explanation:

One of the properties of a parallelogram is that both pairs of opposite angles are congruent. In the given quadrilateral, we can see that two angles are \(50^{\circ}\). Let the other two angles be \(x\). Using the angle - sum property of a quadrilateral (\(360^{\circ}\)), \(2\times50^{\circ}+2x = 360^{\circ}\), which gives \(x = 130^{\circ}\). So both pairs of opposite angles are congruent (\(50^{\circ}\) and \(130^{\circ}\)).
Option B is incorrect because there is no information about the sides being congruent or not. Option C is wrong since the sum of angles of a quadrilateral is \(360^{\circ}\) (here \(2\times50 + 2\times130=100 + 260 = 360^{\circ}\)). Option D is incorrect as the property of a parallelogram is not just one pair of opposite sides congruent and the other pair parallel.

Answer:

A. Yes. It is a parallelogram because both pairs of opposite angles are congruent.