Question

find ∠qrt and ∠trs

Explanation:

Step1: Use the linear - pair property

Since \(\angle QRT\) and \(\angle TRS\) form a linear pair, \(\angle QRT+\angle TRS = 180^{\circ}\). Given \(\angle QRT=3x + 5\) and \(\angle TRS=10x-7\), we have the equation \((3x + 5)+(10x-7)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(3x+10x+5 - 7=180\), which simplifies to \(13x-2 = 180\).

Step3: Solve for \(x\)

Add \(2\) to both sides of the equation: \(13x-2 + 2=180 + 2\), so \(13x=182\). Then divide both sides by \(13\): \(x=\frac{182}{13}=14\).

Step4: Find \(\angle QRT\)

Substitute \(x = 14\) into the expression for \(\angle QRT\): \(\angle QRT=3x+5=3\times14 + 5=42 + 5=47^{\circ}\).

Step5: Find \(\angle TRS\)

Substitute \(x = 14\) into the expression for \(\angle TRS\): \(\angle TRS=10x-7=10\times14-7=140 - 7=133^{\circ}\).

Answer:

\(\angle QRT = 47^{\circ}\), \(\angle TRS=133^{\circ}\)