Question

Let's say you have two line segments, AB and CD which bisect each other at point E. Explain why AC = DB.

Answer 1

Answer:

$AC=DB$

(corresponding parts of congruent triangles are equal)

Step-by-step explanation:

Consider triangles ACE and BDE.

As AB and CD bisect each other at point E,

$AE=BE\\CE=DE$

As AB and CD intersect each other at point E and vertically opposite angles are equal,

∠AEC = ∠BED

So,

ΔAEC ≅ ΔBED       (Using SAS congruence criteria)

(According to SAS congruence criteria, if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle then the two triangles are said to be congruent.)

Hence,

$AC=DB$

(corresponding parts of congruent triangles are equal)

Answer 2

When segments AB and CD bisect each other at point E, the two triangles formed, $\triangle AEC and \triangle BED$, are congruent to each other SAS Congruence Theorem, therefore, AC = DB.

Recall:

• Congruent triangles have equal sides and equal angles that corresponds to each other.

Line segments AB bisecting line segment CD at point E will give us the triangles AEC and BED if we join A to C, and B to D.

Thus, the following information can be deduced from the image of both triangles shown in the attachment below:

$EC \cong ED\\\\AE \cong BE$ (congruent segments)

$\angle AEC \cong \angle BED$ (vertical angles)

From the above stated, $\mathbf{\triangle AEC \cong \triangle BED}$ by the SAS Congruence Theorem.

Since both triangles are congruent, all their corresponding sides would be congruent, therefore AC = DB.

In summary, when segments AB and CD bisect each other at point E, the two triangles formed, $\triangle AEC and \triangle BED$, are congruent to each other SAS Congruence Theorem, therefore, AC = DB.

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