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Part 1
For what values of the variables must ABCD be a​ parallelogram?
5 y degrees5y°
5 x degrees5x°
left parenthesis y plus 72 right parenthesis degrees(y+72)°
left parenthesis 6 x minus 18 right parenthesis degrees(6x−18)°

A
B
C
D

1 answer

To determine the values of \(x\) and \(y\) such that quadrilateral \(ABCD\) is a parallelogram, we need to use the properties of angles in a parallelogram.

In a parallelogram:

  1. Opposite angles are equal.
  2. Adjacent angles are supplementary (their sum is \(180^\circ\)).

Given the angles:

  1. \( \angle A = 5y^\circ \)
  2. \( \angle B = 5x^\circ \)
  3. \( \angle C = (y + 72)^\circ \)
  4. \( \angle D = (6x - 18)^\circ \)

Step 1: Set up equations using properties of a parallelogram

Opposite angles must be equal:

  • \( \angle A = \angle C \)
  • \( \angle B = \angle D \)

This gives us the following equations:

  1. \( 5y = y + 72 \)
  2. \( 5x = 6x - 18 \)

Adjacent angles must be supplementary:

  • \( \angle A + \angle B = 180^\circ \)
  • \( \angle C + \angle D = 180^\circ \)

This gives us the following equations:

  1. \( 5y + 5x = 180 \)
  2. \( (y + 72) + (6x - 18) = 180 \)

Step 2: Solve the equations

From the first two opposite angle equations:

  1. Equation 1: \[ 5y - y = 72 \implies 4y = 72 \implies y = 18 \]

  2. Equation 2: \[ 5x = 6x - 18 \implies -x = -18 \implies x = 18 \]

Now, we have \(y = 18\) and \(x = 18\).

Step 3: Verify by substituting back

Let's use the values of \(x\) and \(y\) to verify the supplementary angle conditions.

  1. Substitute into the supplementary equation (from equation 3): \[ 5(18) + 5(18) = 90 + 90 = 180 \quad \text{(true)} \]

  2. Substitute into the second supplementary equation (from equation 4): \[ (18 + 72) + (6(18) - 18) = 90 + (108 - 18) = 90 + 90 = 180 \quad \text{(true)} \]

Conclusion

The values \(x = 18\) and \(y = 18\) must hold for quadrilateral \(ABCD\) to be a parallelogram.