Question

select the correct answers from each drop - down menu. this construction can be used to prove the angle bisector theorem. using triangle proportionality $\frac{ab}{ad}$ is proportional to. using the isosceles triangle in the construction $ec =$. these lead to the proportion that corresponds to the angle bisector theorem of $\frac{ab}{ad}=$.

Explanation:

Step1: Apply triangle - proportionality

In $\triangle ABD$ and $\triangle ECD$, since $\angle 3=\angle 4$ (vertically - opposite angles) and $\angle 1 = \angle 2$ and $BE\parallel AC$ (construction), by the AA (angle - angle) similarity criterion, $\triangle ABD\sim\triangle ECD$. Then, by the property of similar triangles, $\frac{AB}{AD}=\frac{EC}{DC}$.

Step2: Analyze isosceles triangle

Since $\angle 1=\angle 2$ and $BE\parallel AC$, $\angle 2=\angle 5$ (alternate interior angles) and $\angle 1=\angle 5$. So, $\triangle BCE$ is an isosceles triangle and $EC = BC$.

Step3: Derive angle - bisector theorem

Combining the above two results, we get $\frac{AB}{AD}=\frac{BC}{DC}$ which is the angle - bisector theorem.

Answer:

1. $\frac{EC}{DC}$
2. $BC$
3. $\frac{BC}{DC}$