QUESTION IMAGE
Question
given: \\(\overline{ab} \cong \overline{bc}\\) and \\(\angle abe \cong \angle cbd\\).
prove: \\(\triangle abe \cong \triangle cbd\\).
note: quadrilateral properties are not permitted in this proof.
| step | statement | reason |
|---|---|---|
| 2 | \\(\overline{bd} \cong \overline{bd}\\) | reflexive property |
| 3 | \\(\overline{be} \cong \overline{be}\\) | reflexive property |
| 4 | \\(\overline{bc} \cong \overline{cb}\\) | reflexive property |
| 5 | \\(\overline{ac} \cong \overline{ac}\\) | reflexive property |
| 6 | \\(\overline{ba} \cong \overline{ba}\\) | reflexive property |
| 7 | \\(\overline{ae} \cong \overline{ae}\\) | reflexive property |
| 8 | \\(\overline{ce} \cong \overline{ce}\\) | reflexive property |
| 9 | \\(\overline{da} \cong \overline{da}\\) | reflexive property |
| 10 | \\(\overline{de} \cong \overline{de}\\) | reflexive property |
| 11 | \\(\overline{dc} \cong \overline{dc}\\) | reflexive property |
(there is a dropdown menu labeled \type of statement\ at the bottom.)
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To prove \(\triangle ABE \cong \triangle CBD\), we use the SAS (Side - Angle - Side) congruence criterion.
Step 1: Identify the given congruent sides and angles
We are given that \(\overline{AB}\cong\overline{BC}\) and \(\angle ABE\cong\angle CBD\).
Step 2: Identify the common (reflexive) side
We know that \(\overline{BE}\cong\overline{BE}\) by the Reflexive Property of Congruence (a segment is congruent to itself).
Step 3: Apply the SAS congruence criterion
In \(\triangle ABE\) and \(\triangle CBD\):
- \(\overline{AB}\cong\overline{BC}\) (Given)
- \(\angle ABE\cong\angle CBD\) (Given)
- \(\overline{BE}\cong\overline{BE}\) (Reflexive Property)
By the SAS (Side - Angle - Side) Congruence Postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. So, \(\triangle ABE\cong\triangle CBD\) by SAS.
(Note: The extra reflexive property steps for other segments in the given table are not relevant for this proof as we only need the sides and angle related to \(\triangle ABE\) and \(\triangle CBD\))