Question

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

Explanation:

Step1: Assume it's a quadrilateral

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

Step2: Substitute the given angle expressions

We have \((16x)+m\angle B+(8x + 24)+132 = 360\). Combine like - terms: \(16x+8x+m\angle B+24 + 132=360\), which simplifies to \(24x+m\angle B+156 = 360\). Then, \(m\angle B=360-(24x + 156)=204 - 24x\).
We also know that in a cyclic quadrilateral (if applicable, assuming the context allows this assumption), opposite angles are supplementary. So, \(\angle A+\angle C = 180^{\circ}\) (if it's a cyclic quadrilateral).
Set up the equation \(16x+8x + 24=180\).

Step3: Solve for x

Combine like - terms: \(24x+24 = 180\). Subtract 24 from both sides: \(24x=180 - 24=156\). Then, \(x=\frac{156}{24}=6.5\).

Step4: Find \(m\angle B\)

Substitute \(x = 6.5\) into the expression for \(m\angle B\). \(m\angle B=204-24\times6.5\). First, calculate \(24\times6.5 = 156\). Then, \(m\angle B=204 - 156=48^{\circ}\).

Answer:

\(48\)