Question

given: \\(\overline{ba} \cong \overline{bc}\\) and \\(\overline{ae} \cong \overline{cd}\\).
prove: \\(\overline{be} \cong \overline{bd}\\).
note: quadrilateral properties are not permitted in this proof.
step 1
statement: \\(\overline{ba} \cong \overline{bc}\\), \\(\overline{ae} \cong \overline{cd}\\)
reason: given
(try button and type of statement dropdown)
(image of a triangle with vertices b at the top, a and c at the base, e and d on ac with markings indicating congruent segments)

Explanation:

Step1: Identify the triangles

We have two triangles, $\triangle BAE$ and $\triangle BCD$, and we can also consider $\triangle BEA$ and $\triangle BDC$ or focus on $\triangle BAE$ and $\triangle BCD$ to show congruence. Wait, actually, let's look at $\triangle BAE$ and $\triangle BCD$? No, maybe $\triangle BAE$ and $\triangle BCD$? Wait, no, let's see the sides. We know $BA \cong BC$, $AE \cong CD$, and we need to show $BE \cong BD$. So the triangles involved should be $\triangle BAE$ and $\triangle BCD$? Wait, no, actually, $\triangle BAE$ and $\triangle BCD$? Wait, no, let's check the included angle. Wait, $\angle A$ and $\angle C$: since $BA \cong BC$ and $AE \cong CD$, and if we can show that $\angle A \cong \angle C$, then by SAS congruence, $\triangle BAE \cong \triangle BCD$, which would give $BE \cong BD$. Wait, but how do we know $\angle A \cong \angle C$? Wait, maybe the triangle $BAC$ is isosceles because $BA \cong BC$, so $\angle A \cong \angle C$ (isosceles triangle theorem: if two sides of a triangle are congruent, then the angles opposite them are congruent).

Step2: State $\angle A \cong \angle C$

Since $BA \cong BC$ (given), $\triangle BAC$ is isosceles with base $AC$. Therefore, by the Isosceles Triangle Theorem, $\angle A \cong \angle C$.

Step3: Apply SAS congruence

Now, in $\triangle BAE$ and $\triangle BCD$:

• $BA \cong BC$ (given)
• $\angle A \cong \angle C$ (from Step 2)
• $AE \cong CD$ (given)

Therefore, by the Side - Angle - Side (SAS) Congruence Postulate, $\triangle BAE \cong \triangle BCD$.

Step4: Conclude $BE \cong BD$

Since $\triangle BAE \cong \triangle BCD$, their corresponding sides $BE$ and $BD$ are congruent (Corresponding Parts of Congruent Triangles are Congruent, CPCTC).

Answer:

To prove $\overline{BE} \cong \overline{BD}$, we proceed as follows:

Step 1: State the given information

• $\overline{BA} \cong \overline{BC}$ (Given)
• $\overline{AE} \cong \overline{CD}$ (Given)

Step 2: Show $\angle A \cong \angle C$

Since $\overline{BA} \cong \overline{BC}$, $\triangle BAC$ is isosceles with base $\overline{AC}$. By the Isosceles Triangle Theorem, the angles opposite the congruent sides ($\angle A$ and $\angle C$) are congruent, so $\angle A \cong \angle C$.

Step 3: Prove $\triangle BAE \cong \triangle BCD$

In $\triangle BAE$ and $\triangle BCD$:

• $\overline{BA} \cong \overline{BC}$ (Given)
• $\angle A \cong \angle C$ (From Step 2)
• $\overline{AE} \cong \overline{CD}$ (Given)

By the Side - Angle - Side (SAS) Congruence Postulate, $\triangle BAE \cong \triangle BCD$.

Step 4: Conclude $\overline{BE} \cong \overline{BD}$

Since $\triangle BAE \cong \triangle BCD$, their corresponding sides $\overline{BE}$ and $\overline{BD}$ are congruent (Corresponding Parts of Congruent Triangles are Congruent, CPCTC). Thus, $\overline{BE} \cong \overline{BD}$.