Question

In a parallelogram ABCD, determine the sum of angles C and D.

Answer 1

Hint: There is a property of a parallelogram which says the sum of adjacent interior angles is supplementary (or 180°) and in a parallelogram ABCD angle C and angle D are the adjacent interior angles. So, the sum of angles C and D is 180°.

Complete step-by-step answer:

As we know from the properties of parallel lines that if a transversal is cutting two parallel lines then the sum of interior adjacent angles is 180°. In the below diagram, I have demonstrated the two parallel lines MN and OP and a transversal QR and the two adjacent interior angles.

So, we are going to use the above property of parallel lines in the below parallelogram ABCD:

In a parallelogram, two pair of parallel lines (AD parallel to BC and AB parallel to DC) is present and one pair of parallel lines (AD parallel to BC) crosses with other pair of parallel lines (AB parallel to DC), so one pair of parallel lines (AB and DC) that crosses the other pair of parallel lines (AD and BC) behave as a transversal.
In the question, we are asked to find the sum of angles C and D so we are taking parallel lines AD and BC and DC as the transversal so using the above property of sum of adjacent interior angles we can say angle D and angle C are the adjacent interior angles so their sum should be 180°.
Hence, we have found that the sum of angle C and angle D is 180° (or are supplementary).

Note: We can prove that the sum of adjacent interior angles is 180°.
The following figure contains 2 parallel lines MN and OP and a transversal QR cutting them at T and S respectively.

From the above figure, $\angle TSO=\angle MTS$(corresponding angles)…………… Eq. (1)
And $\angle MTS={{180}^{0}}-\angle QTM$……………… Eq. (2)
$\angle \text{MTS and }\angle \text{TSO}$ are adjacent interior angles that you can see from the figure.
Now, adding eq. (1) and eq. (2) we get:
$\angle TSO+\angle MTS={{180}^{0}}$
Hence, we have proved that the sum of adjacent interior angles is equal to 180°.