Question

in \\( \triangle a b c \\), if \\( \overline{a c} \cong \overline{c b}, m \angle a=(3 x+18)^{\circ}, m \angle b=(7 x - 58)^{\circ} \\), and \\( m \angle c=(2 x - 8)^{\circ} \\), find the value of \\( x \\).

Explanation:

Step1: Use the property of isosceles triangle

Since \(\overline{AC}\cong\overline{CB}\), then \(\angle A=\angle B\) (base angles of an isosceles triangle are equal). But wait, no, actually in \(\triangle ABC\) with \(\overline{AC}\cong\overline{CB}\), \(\angle A=\angle B\) is wrong. Wait, no! Wait, \(\overline{AC}\cong\overline{CB}\), so \(\angle A=\angle B\) is incorrect. Wait, no: in a triangle, if two sides are congruent, the angles opposite those sides are congruent. So if \(\overline{AC}\cong\overline{CB}\), then \(\angle B=\angle A\). Wait, no, \(\overline{AC}\) is opposite \(\angle B\), \(\overline{CB}\) is opposite \(\angle A\). So \(\angle A = \angle B\). Then we use the angle - sum property of a triangle: \(\angle A+\angle B+\angle C=180^{\circ}\). Substitute the given angle expressions: \((3x + 18)+(7x-58)+(2x - 8)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((3x+7x + 2x)+(18-58 - 8)=180\).
\(12x+( - 48)=180\).
\(12x-48 = 180\).

Step3: Solve for \(x\)

Add \(48\) to both sides: \(12x-48 + 48=180 + 48\).
\(12x=228\).
Divide both sides by \(12\): \(x=\frac{228}{12}=19\).

Answer:

\(19\)