In the figure below, given that
𝐴𝐵𝐶𝐷 is a parallelogram, find the measure of angle 𝐴.
In this question, we’re given that
there is a parallelogram, and we need to find one of the angles in this
parallelogram. So, let’s recall what we know about
the angles. We can remember that in a
parallelogram, opposite angles are equal. We’re asked to find the measure of
angle 𝐴. And if we know the angle at 𝐶,
then these would be equal. If we look at the diagram, this
angle 𝐶𝐷𝐹, we don’t know the value of. And if we wanted to work out the
angle 𝐴 directly, we’d need to know this angle at 𝐴𝐷𝐸. But we don’t.
There is another option to work out
the measure of angle 𝐴. And that is, if we know the angle
at 𝐷 or the angle at 𝐵. Let’s see if we can calculate this
angle at 𝐵. Let’s highlight this quadrilateral
within the shape 𝐷𝐸𝐵𝐹. We have a right angle here at angle
𝐷𝐸𝐴. As it lies on the straight line
𝐴𝐵, then we can also say that there is a 90 degree or right angle here at
𝐷𝐸𝐵. So, in this quadrilateral 𝐷𝐸𝐵𝐹,
we have three angles that we know and one that we don’t know. We can use the fact that the angles
in any quadrilateral sum to 360 degrees to find the measure of angle 𝐵.
So, angle 𝐵 is 130 degrees. Because opposite angles in a
parallelogram are equal, then 𝐵 and 𝐷 are both 130 degrees. So, let’s go back to looking at
𝐴𝐵𝐶𝐷. As the sum of the angles in this
parallelogram will be 360 degrees, if we take away two lots of 130 degrees from 360
degrees, then we know that 𝐴 and 𝐶 will add to 100 degrees.
Since we know that 𝐴 and 𝐶 are
equal, then both of these would be 50 degrees. So, the measure of angle 𝐴 is 50