To prove that ∠ADC is a right angle, let's go through the steps systematically based on the given information and hint:

Given Information:

1. AB ≅ AC: Segment AB is congruent to segment AC.
2. AD bisects ∠BAC: Segment AD divides ∠BAC into two equal angles.
3. Prove: ∠ADC is a right angle.
4. Hint: Prove that ΔABD ≅ ΔACD.

Steps to Prove:

Step 1: Show ΔABD ≅ ΔACD

1. AB ≅ AC (Given).

2. AD ≅ AD (Reflexive property of congruence).

3. ∠BAD ≅ ∠CAD (Given that AD bisects ∠BAC).

   Using the Side-Angle-Side (SAS) postulate, we can conclude:

   • ΔABD ≅ ΔACD.

Step 2: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

From the congruence ΔABD ≅ ΔACD, we have:

• ∠BDA ≅ ∠CDA.

Step 3: Prove ∠ADC is a Right Angle

1. In ΔABC, the line segment AD is an angle bisector and divides the isosceles triangle into two congruent triangles.

2. Since ∠BDA and ∠CDA are congruent and supplementary (they form a linear pair along AD), each must measure 90°.

   Therefore, ∠ADC = 90°, proving that it is a right angle.

Would you like further clarification on any of these steps or more details about triangle congruence? Here are five related questions to deepen your understanding:

1. What other methods could be used to prove ΔABD ≅ ΔACD besides SAS?
2. How does the property of isosceles triangles simplify the proof process in this scenario?
3. What is the significance of proving linear pairs are supplementary in geometry proofs?
4. How would this proof change if AD were not an angle bisector?
5. Can this result be generalized to any isosceles triangle with a bisected angle?

Tip: Always check for symmetry and congruence when working with isosceles triangles, as they often simplify geometric proofs!