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 both pairs of opposite angles of the quadrilateral are congruent 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:

$\angle ABC=\angle DAB=90°$and $AB\parallel CD$.

As, $AB\parallel CD$, therefore the angles $\angle ABC$and $\angle BCD$are the angles on the same side of the transversal BC.

Therefore, the sum of the angles $\angle ABC$and $\angle BCD$is $180°$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle ABC+\angle BCD=180°\\ 90°+\angle BCD=180°\\ \angle BCD=180°-90°\\ \angle BCD=90°\end{array}$

Therefore, the measure of the angle $\angle BCD$is $90°$.

As we know that the sum of the angles of a quadrilateral is $360°$.

Therefore, it can be obtained that:

$\begin{array}{c}m\angle ABC+m\angle BCD+m\angle CDA+m\angle DAB=360°\\ 90°+90°+m\angle CDA+90°=360°\\ m\angle CDA+270°=360°\\ m\angle CDA=360°-270°\\ m\angle CDA=90°\end{array}$

Therefore, the measure of the angle $\angle CDA$is $90°$.

Therefore, it can be noticed that $\angle BCD=\angle DAB=90°$and $\angle ABC=\angle ADC=90°$.

Therefore, $\angle BCD\cong \angle DAB$.

Therefore, it can be said that $\angle BCD\cong \angle DAB$and $\angle ABC\cong \angle ADC$.

That implies that both pairs of opposite angles of the quadrilateral are congruent.

Therefore, the given quadrilateral ABCDis a parallelogram.

Step 3. Write the conclusion.

Yes, the given figure ABCD is a parallelogram.

The theorem that applies is that if both pairs of opposite angles of the quadrilateral are congruent then the quadrilateral is a parallelogram.