Question 693777: Find the value of each variable that will allow you to prove the ABCD is a parallelogram. Then find the measure of each angle of the parallelogram.
A=?°
B=(2y-3x)°
C=(x+y)°
D=((5x-y)°
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! When dealing with angles of parallelograms, two things you want to keep in mind are that opposite angles are congruent and adjacent angles are supplementary. In your parallelogram ABCD, angles A and B and angles C and D are adjacent pairs. If you were to draw this out, you would find that these angles are next to each other (probably vertically). These are what you call adjacent angles. Supplementary angles are angles that add up to 180°, so you could say m∠A + m∠B = 180° and m∠C + m∠D = 180°. The two opposite angle pairs are A and C and B and D. If you were to draw parallelogram ABCD, you would find that these angles are diagonal from each other. These are what you call opposite angles. Opposite angles are congruent to each other, so you could say that ∠A ≅ ∠C and
∠B ≅ ∠D. This also means that the measures of these angles are equal, or m∠A = m∠C and m∠B = m∠D.
We need the opposite angle property for this one part. Since we are given a "?" for m∠A (no terms at all), we need to find something about it to work with. Since we know that ∠A ≅ ∠C, we could say this:
∠A ≅ ∠C
m∠A = m∠C
?° = (x+y)°
(x+y)° = (x+y)°.
Now we have something for m∠A. This is namely (x+y)°. Now we could use the supplementary property to find two equations:
Equation 1:
m∠A + m∠B = 180°
(x+y)° + (2y-3x)° = 180°
(3y-2x)° = 180°.
Equation 2:
m∠C + m∠D = 180°
(x+y)° + (5x-y)° = 180°
6x° = 180°
x = 30°.
We have found that x = 30° without having to do any extended substitution. Equation 2 could have easily been an equation with 2 variables, and then we would have had to use elimination or substitution to solve for one variable. In this case, y just cancels out. We still need to use a simple form of substitution. We need to plug x in to find y and the values of the angles. Using equation 1:
(3y-2x)° = 180°
[3y-2(30)]° = 180°
(3y-60)° = 180°
3y° = 240°
y = 80°
Finding the angles' measures:
m∠A, m∠C = (x+y)° = (30+80)° = 110°.
m∠B, m∠D = (2y-3x)°, (5x-y)° = (160-90)°, (150-80)° = 70°, 70°.
We have proven that the opposite angles equal with the values of x and y being 30° and 80° respectively. We have also proved that adjacent pairs are supplementary. Therefore, we have proved ABCD is a parallelogram.
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