Question

in $\triangle abc$, if $overline{ac}congoverline{cb}$, $mangle a=(3x + 18)^{circ}$, $mangle b=(7x - 58)^{circ}$, and $mangle c=(2x - 8)^{circ}$, find the value of $x$ and the measure of each angle.
$x =$
$mangle a =$
$mangle b =$
$mangle c =$

Explanation:

Step1: Use the property of isosceles triangle

Since \(\overline{AC}\cong\overline{CB}\), then \(\angle A\cong\angle B\). So \(3x + 18=7x-58\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(18 = 4x-58\).
Add \(58\) to both sides: \(4x=18 + 58=76\).
Divide both sides by \(4\): \(x = 19\).

Step3: Calculate the measure of each angle

For \(\angle A\): \(m\angle A=(3x + 18)^{\circ}\). Substitute \(x = 19\), \(m\angle A=(3\times19 + 18)^{\circ}=(57+18)^{\circ}=75^{\circ}\).
For \(\angle B\): \(m\angle B=(7x - 58)^{\circ}\). Substitute \(x = 19\), \(m\angle B=(7\times19-58)^{\circ}=(133 - 58)^{\circ}=75^{\circ}\).
For \(\angle C\): \(m\angle C=(2x - 8)^{\circ}\). Substitute \(x = 19\), \(m\angle C=(2\times19-8)^{\circ}=(38 - 8)^{\circ}=30^{\circ}\).

Answer:

\(x = 19\)
\(m\angle A=75^{\circ}\)
\(m\angle B = 75^{\circ}\)
\(m\angle C=30^{\circ}\)