Let's analyze and solve the questions step-by-step from the provided image. Here, I'll address each question individually, starting with question 13 and continuing as needed.

Question 13

Problem Statement: Given a quadrilateral $ABCD$ with the following information:

• $AB = AD$
• $\angle ABD = 90^\circ$
• $\angle C = 15^\circ$
• $\angle DBC = \angle DAC$

We are tasked with finding $\angle BAD$.

Solution:

1. Since $AB = AD$ and $\angle ABD = 90^\circ$, triangle $ABD$ is an isosceles right triangle with $\angle ADB = 45^\circ$ and $\angle BAD = 45^\circ$.

2. With $\angle DBC = \angle DAC$, triangle $DBC$ is also isosceles.

3. Since $\angle C = 15^\circ$ in triangle $ABC$, and $ABCD$ is configured as shown, the specific angle relationships yield $\angle BAD = 45^\circ$ as a result from the right triangle properties and given symmetry.

Therefore, $\angle BAD = 45^\circ$.

Would you like me to continue with other questions, or do you want more details on this solution? Here are some related questions to deepen your understanding:

1. How would the solution change if $AB \neq AD$?
2. What if $\angle ABD$ was not a right angle?
3. How does the isosceles right triangle property impact other angle measurements?
4. Can this configuration exist in a 3D shape, like a tetrahedron?
5. How does knowing one angle in a quadrilateral help determine others?

Tip: In isosceles triangles, the two base angles are always equal, which is useful when calculating unknown angles.