To prove that ΔABC and ΔDBE are congruent, we will show that they are similar by proving that Angle A = Angle D, Angle B = Angle B, and Angle C = Angle E.
Given: AB * BE = CB * BD
Let's first prove that Angle A = Angle D:
Since AB * BE = CB * BD, we can write the proportion AB/CB = BD/BE. This implies that ΔABC ~ ΔDBE by the Side-Angle-Side (SAS) similarity theorem.
Therefore, Angle A = Angle D.
Next, let's prove that Angle B = Angle B:
This is given as both triangles share the common angle B.
Finally, let's prove that Angle C = Angle E:
Since AB * BE = CB * BD, we can write the proportion AB/CB = BD/BE. This implies that ΔABC ~ ΔDBE by the Side-Angle-Side (SAS) similarity theorem.
Therefore, Angle C = Angle E.
Therefore, based on the above proofs, we can conclude that ΔABC ~ ΔDBE.
Write a proof.
Given: AB · BE = CB · BD
Prove: ΔABC ΔDBE
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