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: Use triangle proportionality theorem

By the triangle proportionality theorem (also known as the basic proportionality theorem or Thales' theorem), if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. In this case, since \(AB\parallel CE\), we have \(\frac{AB}{CE}=\frac{AD}{DC}\), so \(\frac{AB}{AD}\) is proportional to \(\frac{CE}{DC}\)

Step2: Use isosceles - triangle property

Since \(\angle 2=\angle 5\) (alternate - interior angles as \(AB\parallel CE\) and \(BE\) is a transversal) and \(\angle 1 = \angle 2\) (given that \(BD\) is an angle bisector), then \(\angle 1=\angle 5\). In \(\triangle BCE\), if \(\angle 1=\angle 5\), then \(EC = BC\) (sides opposite to equal angles in a triangle are equal)

Step3: Substitute \(EC\)

Substituting \(EC = BC\) into \(\frac{AB}{AD}=\frac{CE}{DC}\), we get \(\frac{AB}{AD}=\frac{BC}{DC}\)

Answer:

\(\frac{CE}{DC}\), \(BC\), \(\frac{BC}{DC}\)