Question

challenge given that m∠a=(16x)°, m∠c=(8x + 24)°, and m∠d = 132°, what is m∠b? m∠b =

Explanation:

Step1: Recall sum of angles in a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. So, $m\angle A + m\angle B+m\angle C + m\angle D=360^{\circ}$.

Step2: Substitute the given angle - measures

We know that $m\angle A=(16x)^{\circ}$, $m\angle C=(8x + 24)^{\circ}$, and $m\angle D = 132^{\circ}$. Substituting these into the equation gives $(16x)+m\angle B+(8x + 24)+132 = 360$.

Step3: Combine like - terms

Combine the $x$ terms and the constant terms: $16x+8x+m\angle B+24 + 132=360$, which simplifies to $24x+m\angle B+156 = 360$.

Step4: Isolate the terms with $x$ and $m\angle B$

Subtract 156 from both sides of the equation: $24x+m\angle B=360 - 156$, so $24x+m\angle B=204$.

Step5: Solve for $x$ (assuming it can be determined from other information about the quadrilateral, if it is a special - case quadrilateral. If not, we can express $m\angle B$ in terms of $x$). Let's assume it's a cyclic quadrilateral, and opposite angles are supplementary. $\angle A$ and $\angle C$ are opposite angles in a cyclic quadrilateral, so $m\angle A+m\angle C = 180^{\circ}$.

Set up the equation $16x+8x + 24=180$.

Step6: Combine like - terms in the new equation

$24x+24 = 180$.

Step7: Solve for $x$

Subtract 24 from both sides: $24x=180 - 24=156$. Then $x=\frac{156}{24}=6.5$.

Step8: Substitute $x$ into the sum - of - angles equation

Substitute $x = 6.5$ into $m\angle A + m\angle B+m\angle C + m\angle D=360^{\circ}$.
$m\angle A=16\times6.5 = 104^{\circ}$, $m\angle C=8\times6.5+24=52 + 24=76^{\circ}$.
$104+m\angle B+76+132 = 360$.

Step9: Solve for $m\angle B$

Combine the known angle measures: $104 + 76+132+m\angle B=360$.
$312+m\angle B=360$.
Subtract 312 from both sides: $m\angle B=360 - 312 = 48^{\circ}$.

Answer:

$48$