Question

applying the third corollary to the inscribed angles theorem. the measure of ∠b is (3x - 4)° and the measure of ∠d is (2x - 6)°. what are the measures of angles b and d? m∠b = m∠d = 35 38 70 76

Explanation:

Step1: Recall the property of inscribed - quadrilateral

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary, so \(\angle B+\angle D = 180^{\circ}\).

Step2: Set up the equation

We know that \(m\angle B=(3x - 4)^{\circ}\) and \(m\angle D=(2x - 6)^{\circ}\). Then \((3x - 4)+(2x - 6)=180\).

Step3: Simplify the left - hand side of the equation

Combine like terms: \(3x+2x-4 - 6=180\), which gives \(5x-10 = 180\).

Step4: Solve for \(x\)

Add 10 to both sides of the equation: \(5x=180 + 10=190\). Then divide both sides by 5: \(x=\frac{190}{5}=38\).

Step5: Find the measure of \(\angle B\)

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

Step6: Find the measure of \(\angle D\)

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

Answer:

\(m\angle B = 110\)
\(m\angle D = 70\)