Question

in the diagram below, $overline{ab}congoverline{bd}congoverline{bc}$, and $mangle a = 25^{circ}$. find $mangle c$.

Explanation:

Step1: Identify isosceles - triangle properties

In $\triangle ABD$, since $AB = BD$, then $\angle A=\angle ADB = 25^{\circ}$ (base - angles of an isosceles triangle are equal).

Step2: Calculate $\angle ABD$

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

Step3: Calculate $\angle DBC$

Since $\angle ABC$ is a straight - line angle ($180^{\circ}$), and $\angle ABD = 130^{\circ}$, then $\angle DBC=180^{\circ}-\angle ABD = 180^{\circ}-130^{\circ}=50^{\circ}$.

Step4: Identify isosceles - triangle properties in $\triangle BCD$

In $\triangle BCD$, since $BD = BC$, then $\angle C=\angle BDC$ (base - angles of an isosceles triangle are equal).

Step5: Calculate $\angle C$

Using the angle - sum property of a triangle ($\angle DBC+\angle C+\angle BDC = 180^{\circ}$) in $\triangle BCD$, and since $\angle C=\angle BDC$, we have $2\angle C=180^{\circ}-\angle DBC$. Substituting $\angle DBC = 50^{\circ}$, we get $2\angle C=180^{\circ}-50^{\circ}=130^{\circ}$, so $\angle C = 65^{\circ}$.

Answer:

$65^{\circ}$