Question

in the figure below, \\( \overline{ab} \cong \overline{ad} \\) and \\( \overline{ac} \\) bisects \\( \angle a \\). solve for \\( x \\). then, using that value, find the length of \\( \overline{ac} \\).

Explanation:

Step1: Use the property of congruent triangles

Since \(AB\cong AD\) and \(AC\) bisects \(\angle A\), \(\triangle ABC\cong\triangle ADC\) (by the Side - Angle - Side (SAS) congruence criterion). Then \(BC = DC\).
Set up the equation \(2x + 160=15x + 4\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(160 = 13x+4\).
Subtract \(4\) from both sides: \(156 = 13x\).
Divide both sides by \(13\): \(x = 12\).

Step3: Find the length of \(AC\)

Substitute \(x = 12\) into the expression for \(AC\) (\(AC=11x + 35\)).
\(AC=11\times12+35\).
First, calculate \(11\times12 = 132\).
Then \(AC=132 + 35=167\).

Answer:

\(x = 12\) and the length of \(\overline{AC}\) is \(167\).