Question

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

Explanation:

Step1: Recall the 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

Combining the $x$ terms and the constant terms, we get $(16x+8x)+m\angle B+(24 + 132)=360$, 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: Since we assume this is a cyclic quadrilateral (if not given otherwise, we use the general quadrilateral angle - sum property), and we can also use the fact that in a cyclic quadrilateral, opposite angles are supplementary. If we assume it's a cyclic quadrilateral, $\angle A+\angle C = 180^{\circ}$ (a property of cyclic quadrilaterals). So, $16x+8x + 24=180$.

Step6: Solve for $x$ in the equation $16x+8x + 24 = 180$

Combining like - terms: $24x+24 = 180$. Subtract 24 from both sides: $24x=180 - 24=156$. Then $x=\frac{156}{24}=\frac{13}{2}$.

Step7: Substitute $x$ into the angle - sum equation for the quadrilateral

Substitute $x = \frac{13}{2}$ into $24x+m\angle B = 204$. We have $24\times\frac{13}{2}+m\angle B = 204$.
$12\times13+m\angle B = 204$.
$156+m\angle B = 204$.
Subtract 156 from both sides: $m\angle B=204 - 156 = 48^{\circ}$.

Answer:

$48$