Question

the measure of $\angle$b is $(3x - 4)\degree$ and the measure of $\angle$d is $(2x - 6)\degree$. what are the measures of angles b and d? $m\angle$b = $m\angle$d = 38 76 110 210

Explanation:

Step1: Use the property of cyclic quadrilaterals

In a cyclic quadrilateral, the sum of opposite angles is \(180^{\circ}\). So, \(m\angle B + m\angle D=180^{\circ}\).
Substitute \(m\angle B=(3x - 4)^{\circ}\) and \(m\angle D=(2x - 6)^{\circ}\) into the equation:
\((3x - 4)+(2x - 6)=180\)

Step2: Solve the equation for \(x\)

Combine like - terms:
\(3x+2x-4 - 6=180\)
\(5x-10 = 180\)
Add \(10\) to both sides:
\(5x=180 + 10\)
\(5x=190\)
Divide both sides by \(5\):
\(x=\frac{190}{5}=38\)

Step3: Find \(m\angle B\)

Substitute \(x = 38\) into \(m\angle B=(3x - 4)^{\circ}\)
\(m\angle B=3\times38-4\)
\(m\angle B = 114-4=110^{\circ}\)

Step4: Find \(m\angle D\)

Substitute \(x = 38\) into \(m\angle D=(2x - 6)^{\circ}\)
\(m\angle D=2\times38-6\)
\(m\angle D=76 - 6=70^{\circ}\)

Answer:

\(m\angle B = 110^{\circ}\), \(m\angle D=70^{\circ}\)