To prove that ΔABD ≅ ΔACD, we will use the SSS Congruence Postulate or another appropriate congruence method depending on the available information.

Given:

1. AC ≅ AB (Line AC is congruent to line AB).
2. D is the midpoint of line BC.

To Prove:

ΔABD ≅ ΔACD.

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Proof:

1. AC ≅ AB (Given).

   • This provides one pair of congruent sides.

2. D is the midpoint of line BC (Given).

   • By definition of a midpoint, BD ≅ DC (D divides BC into two equal parts).

3. AD ≅ AD (Reflexive Property).

   • The segment AD is common to both triangles ΔABD and ΔACD.

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Conclusion:

From the above steps, the following pairs of sides are congruent:

• AB ≅ AC (Given).
• BD ≅ DC (Midpoint definition).
• AD ≅ AD (Reflexive Property).

Thus, by the SSS Congruence Postulate, ΔABD ≅ ΔACD.

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Do you need further details or have questions?

5 Related Questions:

1. What is the Reflexive Property in geometry, and how is it used in proofs?
2. How does the definition of a midpoint help establish congruence?
3. Can ΔABD ≅ ΔACD be proved using the SAS Postulate? Why or why not?
4. What are some common congruence postulates/theorems in triangle proofs?
5. How would the proof change if the given information included angles instead of side lengths?

Tip:

Always identify congruent sides or angles shared between triangles early in your proof. This simplifies the logical flow and highlights symmetry in the diagram.