given: $\overline{be}$ bisects $\angle dbc$ and $\overline{be} \parallel \overline{ac}$.
prove: $\overline{ab} \cong \overline{bc}$.
note: quadrilateral properties are not permitted in this proof.
step | statement | reason
1 | $\overline{be}$ bisects $\angle dbc$; $\overline{be} \parallel \overline{ac}$ | given
2 | $\angle dbe \cong \angle ebc$ | an angle bisector divides an angle into two congruent angles
3 | $\angle ebc \cong \angle bca$ | parallel lines cut by a transversal form congruent alternate interior angles
4 | $\angle dbe \cong \angle bca$ | transitive property
5 | $\overline{be} \cong \overline{be}$ | reflexive property
6 | $\angle a \cong \angle a$ | reflexive property
try type of statement

Question

Accepted Answer

# Explanation:
## Step1: Use angle - bisector property
Since $ \overline{BE}$ bisects $ \angle DBC$, by the definition of an angle bisector, $ \angle DBE\cong\angle EBC$.
## Step2: Use parallel - line property
Because $ \overline{BE}\parallel\overline{AC}$, and considering the transversal $ BC$, we have $ \angle EBC\cong\angle BCA$ (alternate - interior angles).
## Step3: Use transitive property of congruence
From $ \angle DBE\cong\angle EBC$ and $ \angle EBC\cong\angle BCA$, by the transitive property of congruence ($a\cong b$ and $b\cong c$ implies $a\cong c$), we get $ \angle DBE\cong\angle BCA$.
## Step4: Consider $ \triangle ABE$ and $ \triangle CBE$
We know that $ \overline{BE}\cong\overline{BE}$ (reflexive property). Also, $ \angle ABE = 180^{\circ}-\angle DBE$ and $ \angle CBE=180^{\circ}-\angle EBC$. Since $ \angle DBE\cong\angle EBC$, then $ \angle ABE\cong\angle CBE$.
## Step5: Use ASA (Angle - Side - Angle) congruence criterion
In $ \triangle ABE$ and $ \triangle CBE$, we have $ \angle ABE\cong\angle CBE$, $ \overline{BE}\cong\overline{BE}$, and $ \angle BAE$ and $ \angle BCE$ (from previous angle congruences). By ASA congruence ($ \angle - \overline{BE}-\angle$), $ \triangle ABE\cong\triangle CBE$.
## Step6: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Since $ \triangle ABE\cong\triangle CBE$, then $ \overline{AB}\cong\overline{BC}$ (CPCTC).

# Answer:
$\overline{AB}\cong\overline{BC}$ is proved as above.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Use angle - bisector property

Step2: Use parallel - line property

Step3: Use transitive property of congruence

Step4: Consider \( \triangle ABE\) and \( \triangle CBE\)

Step5: Use ASA (Angle - Side - Angle) congruence criterion

Step6: Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer: