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 by the substitution property of equality. \\(\frac{ad}{cd}=\frac{eb}{cb}\\) by the triangle pro congruent, so ab = eb. by the substitution pr alternate interior angles are congruent if two parallel lines are cut by a transversal co - interior angles are congruent if two parallel lines are cut by a transversal corresponding sides are congruent if two parallel lines are cut by a transversal corresponding angles are congruent if two parallel lines are cut by a transversal. ∠2≅∠3 ∠3≅∠4 the angles are

Explanation:

Step1: Recall angle - bisector property

Since $\overline{BD}$ bisects $\angle ABC$, we have $\angle 1\cong\angle 2$ by the definition of an angle - bisector.

Step2: Use parallel - line angle relationships

$\overline{EA}\parallel\overline{BD}$. $\angle 2\cong\angle 3$ because alternate interior angles are congruent if two parallel lines are cut by a transversal.

Step3: Use parallel - line angle relationships

$\angle 1\cong\angle 4$ because corresponding angles are congruent if two parallel lines are cut by a transversal.

Step4: Use substitution

By the substitution property, since $\angle 1\cong\angle 2$ and $\angle 1\cong\angle 4$ and $\angle 2\cong\angle 3$, we can show that $\angle 3\cong\angle 4$. Then, in $\triangle ABE$, since $\angle 3\cong\angle 4$, $\triangle ABE$ is isosceles and $AB = EB$.

Step5: Apply triangle - proportionality theorem

By the triangle - proportionality theorem (also known as the basic proportionality theorem or Thales' theorem), in $\triangle CEA$ with $\overline{BD}\parallel\overline{EA}$, we have $\frac{AD}{CD}=\frac{EB}{CB}$.

Step6: Substitute $AB$ for $EB$

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

Answer:

The blanks should be filled as follows:

1. $\angle 1\cong\angle 2$ because $\overline{BD}$ bisects $\angle ABC$.
2. $\angle 2\cong\angle 3$ because alternate interior angles are congruent if two parallel lines are cut by a transversal.
3. $\angle 1\cong\angle 4$ because corresponding angles are congruent if two parallel lines are cut by a transversal.
4. $AB = EB$ because $\angle 3\cong\angle 4$ (sides opposite congruent angles in a triangle are equal).
5. $\frac{AD}{CD}=\frac{EB}{CB}$ by the triangle - proportionality theorem.
6. $\frac{AD}{CD}=\frac{AB}{CB}$ by the substitution property of equality.