Given that 𝐴𝐵𝐶𝐷 is a parallelogram, find the measure of angle 𝐷, the measure of angle 𝐴, and the perimeter of 𝐴𝐵𝐶𝐷.
Before we start this question, it is worth remembering some properties of parallelograms. Firstly, any parallelogram has two pairs of parallel equal length sides. In this question, the sides 𝐴𝐵 and 𝐷𝐶 are parallel and equal in length. As 𝐴𝐵 is equal to 21 centimeters, then 𝐷𝐶 is also equal to 21 centimeters. Likewise, the sides 𝐵𝐶 and 𝐴𝐷 are also parallel and equal in length. This means that the length 𝐴𝐷 is equal to 20 centimeters. The opposite angles in a parallelogram are equal. This means that angle 𝐵 is equal to angle 𝐷, and angle 𝐴 is equal to angle 𝐶. This means that the measure of angle 𝐷 is equal to 72 degrees. This is the first part of our answer.
The angles 𝐴 and 𝐷 are cointerior. This means that they sum or add up to 180 degrees. As angle 𝐷 is 72 degrees, we can calculate angle 𝐴 by subtracting 72 from 180. This is equal to 108. Therefore, the measure of angle 𝐴 is 108 degrees. We have now worked out the first two parts of the question. The final part was to calculate the perimeter of the parallelogram. The perimeter of any shape is the distance around the outside. This can be calculated by adding each of the four lengths. In this case, we need to add 21, 20, 21, and 20. 21 plus 20 is equal to 41. This means that the perimeter is equal to 41 plus 41; this equals 82. So the perimeter of the parallelogram 𝐴𝐵𝐶𝐷 is 82 centimetres. The measure of angle 𝐷 is 72 degrees, angle 𝐴 is 108 degrees, and the perimeter is 82 centimetres.