Question

in $\triangle abc$, $m\angle a = (3x + 1)\degree$, $m\angle b = (x + 14)\degree$, and $m\angle c = (4x + 5)\degree$. what is the value of $x$? a 18 b 19 c 20 d 22

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So, \(m\angle A + m\angle B + m\angle C = 180^\circ\).

Step2: Substitute angle expressions

Substitute \(m\angle A=(3x + 1)^\circ\), \(m\angle B=(x + 14)^\circ\), \(m\angle C=(4x + 5)^\circ\) into the equation:
\((3x + 1)+(x + 14)+(4x + 5)=180\).

Step3: Combine like terms

Combine \(x\)-terms and constants: \(3x + x + 4x + 1 + 14 + 5 = 180\)
\(8x + 20 = 180\).

Step4: Solve for \(x\)

Subtract 20 from both sides: \(8x = 180 - 20 = 160\)
Divide by 8: \(x=\frac{160}{8}=20\).

Answer:

\(x = 20\) (corresponding to option C)