Question

16.) A. Given a parallelogram with angles a, B, C, and D, as in figure 10.90, describe how angle A is related to the other three angles. Your answer should be general and that it would hold for every parallelogram not just the particular parallelogram shown in the figure. B. use the definition of parallelogram and the parallel postulate to explain why the relationships you found in part a hold.

Answer 1

We need to write the Parallel Postulate and the definition for parallelogram:

• A parallelogram is a quadrilateral with two pairs of parallel sides.

• If two lines are cut by a transversal and the corresponding angles are equal, then the lines are parallel.

How are the angles related to each other?

We have corresponding angles with the same color (and also vertical angles). From the definition for parallelogram both sides are parallel. And from it, we can say, using the postulate, we have that the corresponding angles are equals (because lines are parallel).

Then angle blue and red are supplementary (they sum 180), and the angles yellow and green are supplementary too (they also sum 180).

We can also say that:

`\measuredangle a\cong\measuredangle c\text{ }`
`\measuredangle b\cong\measuredangle d`
`m\measuredangle a+m\measuredangle b=180,m\measuredangle c+m\measuredangle d=180`

Additionally, in this way, we can say that the inner angles in a parallelogram sum 360.