Question

Study the markings on each figure and decide whether ABCD must be a parallelogram. If the answer is yes, state the definition or theorem that applies.

Short answer

Yes, the given figure ABCD is a parallelogram.

The theorem that applies is that if one pair of opposite sides are both congruent and parallel then the quadrilateral is a parallelogram.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

From the given diagram it can be noticed that:

$\angle DCA=\angle BAC$and $AD=BC$.

It can be noticed that the angles $\angle DCA$and $\angle BAC$are the alternate interior angles and the transversal is AC.

As, $\angle DCA=\angle BAC$, therefore, it can be said that the alternate interior angles $\angle DCA$and $\angle BAC$are equal.

Therefore, the lines AD and BC are parallel.

Therefore, $AD\parallel BC$.

Therefore, it can be noticed that $AD=BC$and $AD\parallel BC$.

That implies one pair of opposite sides are both congruent and parallel.

Therefore, the given quadrilateral ABCD is a parallelogram.

Step 3. Write the conclusion.

Yes, the given figure ABCD is a parallelogram.

The theorem that applies is that if one pair of opposite sides are both congruent and parallel then the quadrilateral is a parallelogram