Question

what is the measure of ∠a?
□°
what is the measure of ∠b?
□°
(3y + 27)°
(5y - 3)°

Explanation:

Step1: Identify the figure type

The figure is a parallelogram (since opposite sides are parallel, indicated by the arrows). In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)) and opposite angles are equal. Also, consecutive angles like \(\angle A\) and \(\angle C\) are supplementary? Wait, no, in a parallelogram, consecutive angles (adjacent angles) are supplementary. Wait, actually, in a parallelogram, \(\angle A\) and \(\angle B\) are supplementary, \(\angle B\) and \(\angle C\) are supplementary, etc. Also, opposite angles are equal. Wait, looking at the angles: \(\angle A=(5y - 3)^\circ\) and \(\angle C=(3y + 27)^\circ\). In a parallelogram, opposite angles are equal, so \(\angle A=\angle C\)? Wait, no, wait the sides: \(AB\) and \(CD\) are parallel (arrows on \(AB\) and \(CD\)), \(AD\) and \(BC\) are parallel (arrows on \(AD\) and \(BC\)). So \(\angle A\) and \(\angle C\) are opposite angles? Wait, no, in a parallelogram \(ABCD\), vertices are in order, so \(\angle A\) is opposite \(\angle C\), and \(\angle B\) is opposite \(\angle D\). So opposite angles are equal. Therefore, \(\angle A=\angle C\). So set \(5y - 3=3y + 27\).

Step2: Solve for \(y\)

\[

$$\begin{align*} 5y - 3&=3y + 27\\ 5y - 3y&=27 + 3\\ 2y&=30\\ y&=15 \end{align*}$$

\]

Step3: Find \(\angle A\)

Substitute \(y = 15\) into \(\angle A=(5y - 3)^\circ\):
\[
\angle A=5(15)-3=75 - 3 = 72^\circ
\]

Step4: Find \(\angle B\)

In a parallelogram, consecutive angles are supplementary, so \(\angle A+\angle B = 180^\circ\). So \(\angle B=180^\circ-\angle A=180 - 72 = 108^\circ\). Alternatively, we can check with \(\angle C\): \(\angle C=3(15)+27 = 45 + 27 = 72^\circ\), which matches \(\angle A\), so that's correct. Then \(\angle B\) is supplementary to \(\angle A\), so \(180 - 72 = 108\).

Answer:

The measure of \(\angle A\) is \(72^\circ\).
The measure of \(\angle B\) is \(108^\circ\).