Question

Prove a trianglethe given statements are that AB is congruent to BC and line AD is congruent to CEYou have to prove that angle BED is congruent to angle BDEthe subject is geometry

Answer 1

Notice that the triangle ABC is an isosceles triangle, since the sides AB and CB have the same measure. Since congruent sides of an isosceles triangle are opposite to congruent angles, then:

`\angle BAC\cong\angle BCA`

Now, observe the triangles BAD and BCE. We know that:

`\begin{gathered} BA\cong BC \\ \angle BAD\cong\angle BCE \\ AD\cong CE \end{gathered}`

(Note: The angle BAD is the same as BAC, as well as the angle BCE is the same as BCA).

Then, from the SAS congruence criterion, the triangles BAD and BCE are congruent:

`\Delta BAD\cong\Delta BCE`

On the triangles BAD and BCE, the angles BDA and BEC are corresponding angles, since they are opposite to congruent sides.

On the other hand, since Corresponding Parts of Congruent Triangles are Congruent, then:

`\angle BDA\cong\Delta BEC`

Finally, since the angle BDA is the same as BDE and the angle BEC is the same as BED, then:

`\angle BDE\cong BED`

Which completes the proof.