Understanding the Given Conditions
Given the conditions:
- AB = AD
- Angle 1 = Angle 2
- Angle 3 = Angle 4
We need to prove that AP = AQ.
Step-by-Step Proof
1. Triangle Properties
- Since AB = AD, triangles formed (ABP and ADQ) are isosceles.
- The angles opposite to equal sides are equal. Thus, Angle 1 = Angle 2 and Angle 3 = Angle 4 imply symmetry in triangles.
2. Angles Relationships
- Given Angle 1 = Angle 2, the triangles will exhibit congruence due to their equal angles.
- Similarly, Angle 3 = Angle 4 leads to further congruence in the triangles involved.
3. Applying the Law of Sines
- In triangles ABP and ADQ, the Law of Sines applies due to the corresponding angles being equal.
- Thus, we can express the lengths AP and AQ in terms of their opposite angles.
4. Conclusion of Congruence
- Since both triangles share the same angles and have equal sides (AB = AD), by Angle-Side-Angle (ASA) congruence, we conclude that AP = AQ.
Final Verification
- By validating each condition and applying geometric principles, we confirm that AP = AQ holds true under the provided conditions.
This structured reasoning demonstrates the equality of AP and AQ effectively, adhering to the principles of geometry.