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Parallelogram Properties - Problem 1
Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
Recall that a parallelogram has two pairs of consecutive and parallel sides. As a result, the angles within the parallelogram can be seen as angles created by parallel lines and a transversal (the adjacent sides act as transversals). From the properties of same side interior angles, these angles in a parallelogram are supplementary, and opposite angles are congruent.
So, in order to solve for variables in a parallelogram, set the expressions of the angles (a) equal to each other if they are opposite angles, or (b) set an expression equal to 180 - (the expression of the adjacent angle).
You can use the properties of parallelograms to find missing variables. So in this problem we have two variables, we have x and we have y and they’re part of the measures of these angles. So let’s start off by finding x.
Well if I look at x, I have them in consecutive angles and what do I know about consecutive angles in a parallelogram? Well if I look at this diagram right here, I see that consecutive angles are same side interior angles of two parallel lines, so they must be supplementary.
So what I can do is I can say that the quantity of 6x plus 10 plus its same side interior angle, which is 2x plus 10, must add up to 180 degrees. So the key thing here was realizing that we had two parallel lines, we thought of this side as a transversal, so if you want to you can draw in an extension, so these two must be supplementary. So I’m going to combine 6 and 2x and we’re going to get 8x, 10 and 10 is 20, so I’m going to get 180 on the other side.
Now to solve this for x I’m going to subtract 20 from both sides of the equation, 8x equals 160 degrees, divide by 8, divide by 8 and x is 20. So I’m going to write that x is 20.
Now x is not necessarily an angle so I can’t say that this is 20 degrees. X is just 20.
If I were to find y, I could do two things, one I could say that opposite angles in a parallelogram are congruent, or I could say that consecutive angles in a parallelogram are supplementary. I think it’d be easier to say that opposite angles are congruent, so if I know what x is, x we said was 20, so this angle right here is 2 times 20 plus 10 and that’s going to equal y because we know opposite angles must be congruent. So 2 times 20 is 40, 40 plus 10 is y, so we see that y must be 50. Now y is an angle measure, so I’m going to write that y is 50 degrees.
Again the key thing here was realizing that same side interior angles are supplementary and that opposite angles in a parallelogram must be congruent.
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