Question

1. a pizza slice is bisected. determine the value of x, if m∠pza = 84° and m∠mza=(3x + 6)°. a. 12 b. 16 c. 26 d. 54 2. given m∠bca=(4(x + 3))° and m∠dce=(6(x - 7))°, determine each of the following. part a: determine the value of x. part b: what is the measure of ∠bca?

Explanation:

Step1: Use angle - bisector property

Since the pizza slice is bisected, \(m\angle PZA = 2m\angle MZA\). Given \(m\angle PZA=84^{\circ}\) and \(m\angle MZA=(3x + 6)^{\circ}\), we have the equation \(84 = 2(3x + 6)\).

Step2: Expand the right - hand side

Expand \(2(3x + 6)\) to get \(6x+12\). So the equation becomes \(84=6x + 12\).

Step3: Solve for \(x\)

Subtract 12 from both sides: \(84−12=6x\), which simplifies to \(72 = 6x\). Then divide both sides by 6: \(x=\frac{72}{6}=12\).

Step1: Use vertical - angle property

\(\angle BCA\) and \(\angle DCE\) are vertical angles, so \(m\angle BCA=m\angle DCE\). Set up the equation \(4(x + 3)=6(x - 7)\).

Step2: Expand both sides

Expand \(4(x + 3)\) to \(4x+12\) and \(6(x - 7)\) to \(6x-42\). The equation is \(4x + 12=6x-42\).

Step3: Isolate \(x\) terms

Subtract \(4x\) from both sides: \(12=6x-4x - 42\), which simplifies to \(12 = 2x-42\).

Step4: Solve for \(x\)

Add 42 to both sides: \(12 + 42=2x\), so \(54 = 2x\). Divide both sides by 2: \(x = 27\).

Step1: Substitute \(x\) into \(m\angle BCA\) formula

We know \(m\angle BCA=4(x + 3)\) and \(x = 27\). Substitute \(x\) into the formula: \(m\angle BCA=4(27 + 3)\).

Step2: Calculate the value

First, \(27+3 = 30\). Then \(4\times30=120^{\circ}\).

Answer:

A. 12