Question

what is the measure of ∠a? what is the measure of ∠d?

Explanation:

1. First, note the property of a parallelogram:

• In a parallelogram, adjacent - angles are supplementary, that is, the sum of adjacent angles is \(180^{\circ}\). In parallelogram \(ABCD\), \(\angle A+\angle B = 180^{\circ}\) and \(\angle A\) and \(\angle C\) are opposite angles (opposite angles of a parallelogram are equal), and \(\angle B\) and \(\angle D\) are opposite angles. Also, we can use the fact that \(\angle A+\angle B=180^{\circ}\), where \(\angle A=(5y - 3)^{\circ}\) and \(\angle B=(3y + 27)^{\circ}\).
• Set up the equation: \((5y - 3)+(3y + 27)=180\).
• Combine like - terms: \(5y+3y-3 + 27=180\), which simplifies to \(8y+24 = 180\).
• Subtract 24 from both sides: \(8y=180 - 24=156\).
• Divide both sides by 8: \(y=\frac{156}{8}=19.5\).

1. Then, find the measure of \(\angle A\):

• Substitute \(y = 19.5\) into the expression for \(\angle A\): \(\angle A=(5y - 3)^{\circ}\).
• \(\angle A=5\times19.5-3=97.5 - 3=94.5^{\circ}\).

1. Next, find the measure of \(\angle D\):

• Since \(\angle D\) and \(\angle B\) are opposite angles in a parallelogram, \(\angle D=\angle B\).
• Substitute \(y = 19.5\) into the expression for \(\angle B\): \(\angle B=(3y + 27)^{\circ}\).
• \(\angle B=3\times19.5+27=58.5+27 = 85.5^{\circ}\), so \(\angle D = 85.5^{\circ}\).

Answer:

Measure of \(\angle A\): \(94.5\)
Measure of \(\angle D\): \(85.5\)