Question

use efficient methods find ( mangle abc ).

Explanation:

Step1: Use the triangle congruence

Since \(AD = DC\) and \(AC = CB\) (marked as equal), and \(\angle ADC = 92^{\circ}\). First, find \(\angle ACD\). In \(\triangle ADC\), \(\angle DAC=\angle DCA\) (isosceles triangle property). Let \(\angle DAC = \angle DCA=x\). Then \(x + x+92^{\circ}=180^{\circ}\) (sum of angles in a triangle). So \(2x=180^{\circ}-92^{\circ}=88^{\circ}\), \(x = 44^{\circ}\). Then \(\angle ACB = 180^{\circ}-\angle ACD=180^{\circ}- 44^{\circ}=136^{\circ}\).

Step2: Use the isosceles triangle property again

In \(\triangle ACB\), \(AC = CB\), so \(\angle CAB=\angle CBA\). Let \(\angle CAB=\angle CBA = y\). Then \(y + y+136^{\circ}=180^{\circ}\) (sum of angles in a triangle). \(2y=180^{\circ}-136^{\circ}=44^{\circ}\), \(y = 22^{\circ}\).

Answer:

\(22\)