Question

diagram below, ( overline{ab} cong overline{bd} cong overline{bc} ), and ( mangle a = 25^{circ} ). find ( mangle c ).

Explanation:

Step1: Find the measure of ∠ADB

Since \( \overline{AB}\cong\overline{BD}\), in \(\triangle ABD\), \(\angle A=\angle ADB = 25^{\circ}\) (base angles of an isosceles triangle are equal).

Step2: Find the measure of ∠ABD

Using the angle - sum property of a triangle (\(\angle A+\angle ADB+\angle ABD = 180^{\circ}\)), we have \(\angle ABD=180^{\circ}-\angle A - \angle ADB\). Substituting \(\angle A = 25^{\circ}\) and \(\angle ADB = 25^{\circ}\), we get \(\angle ABD=180^{\circ}-25^{\circ}-25^{\circ}=130^{\circ}\).

Step3: Find the measure of ∠CBD

Since \(\angle ABD+\angle CBD = 180^{\circ}\) (linear pair), \(\angle CBD = 180^{\circ}-\angle ABD\). Substituting \(\angle ABD = 130^{\circ}\), we get \(\angle CBD=50^{\circ}\).

Step4: Find the measure of ∠C

Since \( \overline{BD}\cong\overline{BC}\), in \(\triangle BCD\), \(\angle C=\angle BDC\) (base angles of an isosceles triangle are equal). Using the angle - sum property of a triangle (\(\angle CBD+\angle C+\angle BDC = 180^{\circ}\)), and since \(\angle C=\angle BDC\), we have \(2\angle C=180^{\circ}-\angle CBD\). Substituting \(\angle CBD = 50^{\circ}\), we get \(2\angle C=180^{\circ}-50^{\circ}=130^{\circ}\), then \(\angle C = 65^{\circ}\).

Answer:

\(65^{\circ}\)