Question

24.
q (w - 16)° t
w° 2(w - 18)°
r s
a. w
b. m∠qrt
c. m∠trs

Explanation:

Step1: Set up the equation

Since the sum of angles on a straight line is \(180^{\circ}\), we have \(w+(w - 16)+2(w - 18)=180\).

Step2: Simplify the left - hand side of the equation

Expand the terms: \(w+w - 16+2w-36 = 180\).
Combine like terms: \((w + w+2w)+(-16 - 36)=180\), which gives \(4w-52 = 180\).

Step3: Solve for \(w\)

Add \(52\) to both sides of the equation: \(4w-52 + 52=180 + 52\), so \(4w=232\).
Divide both sides by \(4\): \(w=\frac{232}{4}=58\).

Step4: Find \(m\angle QRT\)

Substitute \(w = 58\) into the expression for \(\angle QRT\). The measure of \(\angle QRT=(w - 16)^{\circ}\).
So \(m\angle QRT=(58 - 16)^{\circ}=42^{\circ}\).

Step5: Find \(m\angle TRS\)

Substitute \(w = 58\) into the expression for \(\angle TRS\). The measure of \(\angle TRS = 2(w - 18)^{\circ}\).
First, calculate \(w-18=58 - 18 = 40\). Then \(m\angle TRS=2\times40^{\circ}=80^{\circ}\).

Answer:

a. \(w = 58\)
b. \(m\angle QRT=42^{\circ}\)
c. \(m\angle TRS = 80^{\circ}\)