Question

select the correct answer from each drop - down menu. in △abc, side bc is extended to point e. when connected to vertex a, segment ea is parallel to segment bd. in this construction, you are given that bd bisects ∠abc. prove: $\frac{ad}{cd}=\frac{ab}{cb}$. complete the paragraph proof. because of the corresponding angles theorem. ∠1≅∠4 because since bd bisects ∠abc and the two angles created on each side of the bisector at point b are equal. ∠2≅∠3 by the substitution property of equality. $\frac{ad}{cd}=\frac{eb}{cb}$ by the triangle proportionality theorem. if two angles in a triangle are congruent, the sides opposite the angles are congruent, so ab = eb. by the substitution property of equality.

Explanation:

Step1: Use angle - bisector and parallel - line properties

Since $\overline{BD}$ bisects $\angle ABC$, $\angle1=\angle2$. And because $EA\parallel BD$, by the corresponding - angles theorem, $\angle1 = \angle4$ and $\angle2=\angle3$.

Step2: Apply isosceles - triangle property

Since $\angle1=\angle2$ and $\angle1 = \angle4$, $\angle2=\angle4$. In $\triangle ABE$, if two angles ($\angle2$ and $\angle4$) are congruent, then the sides opposite them are congruent, so $AB = EB$ by the isosceles - triangle theorem (if two angles of a triangle are congruent, the sides opposite those angles are congruent).

Step3: Use triangle - proportionality theorem

By the triangle proportionality theorem, since $EA\parallel BD$ in $\triangle BCE$, we have $\frac{AD}{CD}=\frac{EB}{CB}$.

Step4: Substitute $AB$ for $EB$

Since $AB = EB$, by the substitution property of equality, $\frac{AD}{CD}=\frac{AB}{CB}$.

Answer:

The proof is completed as above to show $\frac{AD}{CD}=\frac{AB}{CB}$ using angle - bisector, parallel - line, isosceles - triangle, and triangle - proportionality theorems along with substitution property.