Question

prove that ∠abc is a right angle. given: ad ≅ bd ≅ cd prove: m∠abc = 90 complete the proof below. statement ad ≅ bd ≅ cd △adb and △cdb are isosceles. reason given

Explanation:

Step1: Recall isosceles - triangle property

In $\triangle ADB$, since $AD\cong BD$, $\angle A=\angle ABD$ (Base - angles of an isosceles triangle are equal).

Step2: Recall isosceles - triangle property for another triangle

In $\triangle CDB$, since $BD\cong CD$, $\angle C=\angle CBD$ (Base - angles of an isosceles triangle are equal).

Step3: Use angle - sum property of a triangle

In $\triangle ABC$, $\angle A+\angle ABC+\angle C = 180^{\circ}$. And $\angle ABC=\angle ABD+\angle CBD$.
Since $\angle A=\angle ABD$ and $\angle C=\angle CBD$, we can substitute to get $2\angle ABD + 2\angle CBD=180^{\circ}$.

Step4: Simplify the equation

Dividing both sides of $2\angle ABD + 2\angle CBD = 180^{\circ}$ by 2 gives $\angle ABD+\angle CBD = 90^{\circ}$.
Since $\angle ABC=\angle ABD+\angle CBD$, then $\angle ABC = 90^{\circ}$.

Answer:

We have proved that $\angle ABC$ is a right - angle.