Answer:
Explanation:
To prove that two triangles are congruent, we need to show that they have the same three sides and the same three angles.
Since ∠BAC ≅ ∠BCA, we know that these two triangles have at least one congruent angle. We are given that CD bisects ∠BCA and AE bisects ∠BAC, so these two triangles also have two pairs of congruent angles.
Therefore, we just need to show that the three sides of the triangles are congruent. Since CD bisects ∠BCA and AE bisects ∠BAC, we can apply the Angle Bisector Theorem to triangles ADC and CEA. This theorem states that if a line bisects an angle in a triangle, then it also divides the opposite side into segments that are proportional to the other two sides of the triangle.
Since the Angle Bisector Theorem applies to both triangles ADC and CEA, we can conclude that the ratios of the sides opposite the bisected angles are equal. Specifically, we have:
(AD/DC) = (CE/EA)
Since these ratios are equal, the sides AD, DC, CE, and EA are all proportional. This means that if we can show that two of these sides are congruent, then all four sides are congruent.
Therefore, it remains to show that either AD is congruent to CE or DC is congruent to EA. Since ∠BAC ≅ ∠BCA and ∠BAC and ∠BCA are opposite angles in triangles ADC and CEA, respectively, we can apply the AA (Angle-Angle) Congruence Postulate to these two triangles. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are congruent.
Therefore, we can conclude that triangles ADC and CEA are congruent, which means that quadrilateral ADCE is actually a parallelogram. This means that opposite sides AD and CE are congruent, as well as opposite sides DC and EA.
Therefore, we have shown that all three sides of triangles ADC and CEA are congruent, as well as all three angles. This means that triangles ADC and CEA are congruent by the ASA (Angle-Side-Angle) Congruence Postulate.
Thus, we have proven that ⊿ADC ≅ ⊿CEA.