Question

in the diagram below, $overline{ab} cong overline{bd} cong overline{bc}$, and $mangle c = 65^circ$. find $mangle adc$.

Explanation:

Step1: Identify isosceles triangle $\triangle BDC$

Since $\overline{BD} \cong \overline{BC}$, $\triangle BDC$ is isosceles with $\angle C = \angle BDC = 65^\circ$.

Step2: Calculate $\angle DBC$

Sum of angles in a triangle is $180^\circ$.
$\angle DBC = 180^\circ - 2\times65^\circ = 50^\circ$

Step3: Find $\angle ABD$

$\angle DBC$ and $\angle ABD$ are supplementary.
$\angle ABD = 180^\circ - 50^\circ = 130^\circ$

Step4: Identify isosceles triangle $\triangle ABD$

Since $\overline{AB} \cong \overline{BD}$, $\triangle ABD$ is isosceles with $\angle A = \angle ADB$.

Step5: Calculate $\angle ADB$

Sum of angles in a triangle is $180^\circ$.
$\angle ADB = \frac{180^\circ - 130^\circ}{2} = 25^\circ$

Step6: Calculate $\angle ADC$

$\angle ADC = \angle BDC + \angle ADB$
$\angle ADC = 65^\circ + 25^\circ$

Answer:

$90^\circ$