We are given a quadrilateral $ABCD$ where the following conditions hold:

• $AB = BC = AD$
• $\angle DAB = 90^\circ$
• $\angle ABC = 60^\circ$

Step-by-Step Solution:

Step 1: Identify the geometry of quadrilateral $ABCD$

• Since $AB = AD$, triangle $ABD$ is isosceles, and with $\angle DAB = 90^\circ$, this makes $\triangle ABD$ a right isosceles triangle.
• In a right isosceles triangle, the two base angles are equal. So, $\angle ABD = \angle ADB = 45^\circ$.

Step 2: Use angle sum in triangle $ABC$

• $\triangle ABC$ has $AB = BC$, so it is also an isosceles triangle.
• From the given, $\angle ABC = 60^\circ$, so the remaining two angles in $\triangle ABC$ must sum to $120^\circ$ (since the angle sum in any triangle is $180^\circ$).
• Let $\angle BAC = \angle ACB = x$. Then, $2x + 60^\circ = 180^\circ$ $2x = 120^\circ \implies x = 60^\circ$ Hence, $\angle BAC = \angle ACB = 60^\circ$.

Step 3: Find angle $BCD$

• Now, consider quadrilateral $ABCD$. The sum of the interior angles of any quadrilateral is $360^\circ$. Therefore, we can write: $\angle DAB + \angle ABC + \angle BCD + \angle ADB = 360^\circ$ Substituting the known angles: $90^\circ + 60^\circ + \angle BCD + 45^\circ = 360^\circ$ Simplifying this: $195^\circ + \angle BCD = 360^\circ$ $\angle BCD = 360^\circ - 195^\circ = 165^\circ$

Thus, the value of $\angle BCD$ is $165^\circ$.

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Would you like more details on any of these steps or have any questions?

Here are 5 related questions for further practice:

1. Find the area of triangle $ABD$.
2. If $AB = 5$ units, find the length of diagonal $BD$.
3. What type of quadrilateral is $ABCD$ based on the given information?
4. How does the sum of the angles in a quadrilateral help in solving problems?
5. Can you find the perimeter of $ABCD$ given $AB = 5$ units?

Tip: In any isosceles triangle, the base angles are always equal, which can simplify the calculation of unknown angles.